Tootfinder

Capacity Provisioning Motivated Online Non-Convex Optimization Problem with Memory and Switching Cost
Rahul Vaze, Jayakrishnan Nair
https://arxiv.org/abs/2403.17480

Capacity Provisioning Motivated Online Non-Convex Optimization Problem with Memory and Switching Cost
An online non-convex optimization problem is considered where the goal is to minimize the flow time (total delay) of a set of jobs by modulating the number of active servers, but with a switching cost associated with changing the number of active servers over time. Each job can be processed by at most one fixed speed server at any time. Compared to the usual online convex optimization (OCO) problem with switching cost, the objective function considered is non-convex and more importantly, at eac…

This https://arxiv.org/abs/2305.08260 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Siciak-Zakharyuta theorem
The classical Siciak-Zakharyuta theorem states that the Siciak-Zakharyuta function $V_{E}$ of a subset $E$ of $\mathbb C^n$, also called a pluricomplex Green function or global exremal function of $E$, equals the logarithm of the Siciak function $Φ_E$ if $E$ is compact. The Siciak-Zakharyuta function is defined as the upper envelope of functions in the Lelong class that are negative on $E$, and the Siciak function is the upper envelope of $m$-th roots of polynomials $p$ in $\mathcal{P}_m(\math…

This https://arxiv.org/abs/2403.06708 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Tikhonov Regularization for Stochastic Non-Smooth Convex Optimization in Hilbert Spaces
To solve non-smooth convex optimization problems with a noisy gradient input, we analyze the global behavior of subgradient-like flows under stochastic errors. The objective function is composite, being equal to the sum of two convex functions, one being differentiable and the other potentially non-smooth. We then use stochastic differential inclusions where the drift term is minus the subgradient of the objective function, and the diffusion term is either bounded or square-integrable. In this …

Optimal convex $M$-estimation via score matching
Oliver Y. Feng, Yu-Chun Kao, Min Xu, Richard J. Samworth
https://arxiv.org/abs/2403.16688 https://<…

Optimal convex $M$-estimation via score matching
In the context of linear regression, we construct a data-driven convex loss function with respect to which empirical risk minimisation yields optimal asymptotic variance in the downstream estimation of the regression coefficients. Our semiparametric approach targets the best decreasing approximation of the derivative of the log-density of the noise distribution. At the population level, this fitting process is a nonparametric extension of score matching, corresponding to a log-concave projectio…

Generalized convex functions and their applications in optimality conditions
Mohammad Hossein Alizadeh
https://arxiv.org/abs/2402.15597 https://

Generalized convex functions and their applications in optimality conditions
We introduce and study the notion of (e,y)-conjugate for a proper and e-convex function in locally convex spaces, which is an extension of the concept of the conjugate. The mutual relationships between the concepts of (e,y)-conjugacy and e-subdifferential are presented. Moreover, some applications of these notions in optimization are established.

This https://arxiv.org/abs/2211.00131 has been replaced.
link: https://scholar.google.com/scholar?q=a

New Concept for the Value Function of Prospect Theory
In the prospect theory, value function is typically concave for gains, commonly convex for losses, with losses usually having a steeper slope than gains. The neural system largely differs from the loss and gains sides. Five new studies on neurons related to this issue have examined neuronal responses to losses, gains, and reference points. This study investigates a new concept of the value function. A value function with a neuronal cusp may show variations and behavior cusps with catastrophe wh…

Moving higher-order Taylor approximations method for smooth constrained minimization problems
Yassine Nabou, Ion Necoara
https://arxiv.org/abs/2402.15022 h…

Moving higher-order Taylor approximations method for smooth constrained minimization problems
In this paper we develop a higher-order method for solving composite (non)convex minimization problems with smooth (non)convex functional constraints. At each iteration our method approximates the smooth part of the objective function and of the constraints by higher-order Taylor approximations, leading to a moving Taylor approximation method (MTA). We present convergence guarantees for MTA algorithm for both, nonconvex and convex problems. In particular, when the objective and the constraints …

This https://arxiv.org/abs/2312.07159 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csIT_…

Rate-Splitting Multiple Access for Semantic-Aware Networks: an Age of Incorrect Information Perspective
In this letter, we design a downlink multi-user communication framework based on Rate-Splitting Multiple Access (RSMA) for semantic-aware networks. First, we formulate an optimization problem to obtain the optimal user scheduling, precoding, and power allocation schemes jointly. We consider the metric Age of Incorrect Information (AoII) in the objective function of the formulated problem to maximize the freshness of the overall information to be transmitted. Using big-M and Successive Convex Ap…

This https://arxiv.org/abs/2311.03481 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

$\mathbf{C^2}$-Lusin approximation of strongly convex functions
We prove that if $u:\mathbb{R}^n\to\mathbb{R}$ is strongly convex, then for every $\varepsilon>0$ there is a strongly convex function $v\in C^2(\mathbb{R}^n)$ such that $|\{u\neq v\}|<\varepsilon$ and $\Vert u-v\Vert_\infty<\varepsilon$.

This https://arxiv.org/abs/2310.17350 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A mixed FEM for a time-fractional Fokker-Planck model
We propose and analyze a mixed finite element method for the spatial approximation of a time-fractional Fokker--Planck equation in a convex polyhedral domain, where the given driving force is a function of space. Taking into account the limited smoothing properties of the model, and considering an appropriate splitting of the errors, we employed a sequence of clever energy arguments to show optimal convergence rates with respect to both approximation properties and regularity results. In partic…

This https://arxiv.org/abs/2111.09765 has been replaced.
link: https://scholar.google.com/scholar?q=a

Boundary $C^{2, α}$ Regularity for the Oblique Boundary Value Problem of Monge-Ampère Equations
We study the good shape property of boundary sections of convex solutions of the oblique boundary value problem for Monge-Ampère equations $$\det D^2u =f(x) \text{ in } Ω, \quad D_βu = ϕ(x) \text{ on } \partial Ω.$$ In the two-dimensional case, we prove the global $C^{2,α}$ estimate for the solution. When the dimension $n \geq 3$, we show that this estimate still holds if the solution is bounded from above by a quadratic function in the tangent direction. We also obtain an existence resul…

This https://arxiv.org/abs/2311.00957 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

An equivalent reformulation and multi-proximity gradient algorithms for a class of nonsmooth fractional programming
In this paper, we consider a class of structured fractional programs, where the numerator part is the sum of a block-separable (possibly nonsmooth nonconvex) function and a locally Lipschitz differentiable (possibly nonconvex) function, while the denominator is a convex (possibly nonsmooth) function. We first present a novel reformulation for the original problem and show the relationship between optimal solutions, critical points and KL exponents of these two problems. Inspired by the reformul…

Lefschetz operators on convex valuations
Leo Brauner, Georg C. Hofst\"atter, Oscar Ortega-Moreno
https://arxiv.org/abs/2402.14731 https://

Lefschetz operators on convex valuations
We investigate the action of Alesker's Lefschetz operators on translation invariant valuations on convex bodies. For scalar valued valuations, we describe this action on the level of Klain-Schneider functions by a Radon type transform, generalizing a result by Schuster and Wannerer. In the case of rotationally equivariant Minkowski valuations, the Lefschetz operators act on the generating function as a convolution transform. We show that the convolution kernel satisfies a Legendre type differen…

This https://arxiv.org/abs/2311.08321 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Peak Estimation of Rational Systems using Convex Optimization
This paper presents algorithms that upper-bound the peak value of a state function along trajectories of a continuous-time system with rational dynamics. The finite-dimensional but nonconvex peak estimation problem is cast as a convex infinite-dimensional linear program in occupation measures. This infinite-dimensional program is then truncated into finite-dimensions using the moment-Sum-of-Squares (SOS) hierarchy of semidefinite programs. Prior work on treating rational dynamics using the mome…

This https://arxiv.org/abs/2112.03540 has been replaced.
link: https://scholar.google.com/scholar?q=a

A theory of optimal convex regularization for low-dimensional recovery
We consider the problem of recovering elements of a low-dimensional model from under-determined linear measurements. To perform recovery, we consider the minimization of a convex regularizer subject to a data fit constraint. Given a model, we ask ourselves what is the "best" convex regularizer to perform its recovery. To answer this question, we define an optimal regularizer as a function that maximizes a compliance measure with respect to the model. We introduce and study several notions of co…

This https://arxiv.org/abs/2201.10017 has been replaced.
link: https://scholar.google.com/scholar?q=a

Online Convex Optimization Using Coordinate Descent Algorithms
This paper considers the problem of online optimization where the objective function is time-varying. In particular, we extend coordinate descent type algorithms to the online case, where the objective function varies after a finite number of iterations of the algorithm. Instead of solving the problem exactly at each time step, we only apply a finite number of iterations at each time step. Commonly used notions of regret are used to measure the performance of the online algorithm. Moreover, coo…

This https://arxiv.org/abs/2306.04732 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csRO_…

Online Multi-Contact Receding Horizon Planning via Value Function Approximation
Planning multi-contact motions in a receding horizon fashion requires a value function to guide the planning with respect to the future, e.g., building momentum to traverse large obstacles. Traditionally, the value function is approximated by computing trajectories in a prediction horizon (never executed) that foresees the future beyond the execution horizon. However, given the non-convex dynamics of multi-contact motions, this approach is computationally expensive. To enable online Receding Ho…

Convexity, Fourier transforms, and lattice point discrepancy
Michael Greenblatt
https://arxiv.org/abs/2402.16636 https://arxiv.org/pd…

Convexity, Fourier transforms, and lattice point discrepancy
In a well-known paper by Bruna, Nagel and Wainger [BNW], Fourier transform decay estimates were proved for smooth hypersurfaces of finite line type bounding a convex domain. In this paper, we generalize their results in the following ways. First, for a surface that is locally the graph of a convex real analytic function, we show that a natural analogue holds even when the surface in question is not of finite line type. Secondly, we show a result for a general surface that is locally the graph o…

This https://arxiv.org/abs/2211.00131 has been replaced.
link: https://scholar.google.com/scholar?q=a

New Concept for the Value Function of Prospect Theory
In the prospect theory, value function is typically concave for gains, commonly convex for losses, with losses usually having a steeper slope than gains. The neural system largely differs from the loss and gains sides. Five new studies on neurons related to this issue have examined neuronal responses to losses, gains, and reference points. This study investigates a new concept of the value function. A value function with a neuronal cusp may show variations and behavior cusps with catastrophe wh…

This https://arxiv.org/abs/2309.06429 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_sta…

Efficient Inference on High-Dimensional Linear Models with Missing Outcomes
This paper is concerned with inference on the regression function of a high-dimensional linear model when outcomes are missing at random. We propose an estimator which combines a Lasso pilot estimate of the regression function with a bias correction term based on the weighted residuals of the Lasso regression. The weights depend on estimates of the missingness probabilities (propensity scores) and solve a convex optimization program that trades off bias and variance optimally. Provided that the…

Combinatorics of generalized parking-function polytopes
Margaret M. Bayer, Steffen Borgwardt, Teressa Chambers, Spencer Daugherty, Aleyah Dawkins, Danai Deligeorgaki, Hsin-Chieh Liao, Tyrrell McAllister, Angela Morrison, Garrett Nelson, Andr\'es R. Vindas-Mel\'endez
https://arxiv.org/abs/2403.07387

Combinatorics of generalized parking-function polytopes
For $\mathbf{b}=(b_1,\dots,b_n)\in \mathbb{Z}_{>0}^n$, a $\mathbf{b}$-parking function is defined to be a sequence $(β_1,\dots,β_n)$ of positive integers whose nondecreasing rearrangement $β'_1\leq β'_2\leq \cdots \leq β'_n$ satisfies $β'_i\leq b_1+\cdots + b_i$. The $\mathbf{b}$-parking-function polytope $\mathfrak{X}_n(\mathbf{b})$ is the convex hull of all $\mathbf{b}$-parking functions of length $n$ in $\mathbb{R}^n$. Geometric properties of $\mathfrak{X}_n(\mathbf{b})$ were previousl…

This https://arxiv.org/abs/2312.05166 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Multi-Agent Reinforcement Learning via Distributed MPC as a Function Approximator
This paper presents a novel approach to multi-agent reinforcement learning (RL) for linear systems with convex polytopic constraints. Existing work on RL has demonstrated the use of model predictive control (MPC) as a function approximator for the policy and value functions. The current paper is the first work to extend this idea to the multi-agent setting. We propose the use of a distributed MPC scheme as a function approximator, with a structure allowing for distributed learning and deploymen…

Invertibility of local geodesic transverse and mixed ray transforms II: higher order tensors
Gunther Uhlmann, Jian Zhai
https://arxiv.org/abs/2402.12640 ht…

Invertibility of local geodesic transverse and mixed ray transforms II: higher order tensors
Consider a compact Riemannian manifold in dimension $n$ with strictly convex boundary. We show the local invertibility near a boundary point of the transverse ray transform of $2$ tensors for $n\geq 3$ and the mixed ray transform of $2+2$ tensors for $n=3$. When the manifold admits a strictly convex function, this local invertibility result leads to global invertibility.

This https://arxiv.org/abs/2404.12159 has been replaced.
initial toot: https://mastoxiv.page/@…

Localization in boundary-driven lattice models
Several systems may display an equilibrium condensation transition, where a finite fraction of a conserved quantity is spatially localized. The presence of two conservation laws may induce the emergence of such transition in an out-of-equilibrium setup, where boundaries are attached to two different and subcritical heat baths. We study this phenomenon in a class of stochastic lattice models, where the local energy is a general convex function of the local mass, mass and energy being both global…

One-Bit Quantization and Sparsification for Multiclass Linear Classification via Regularized Regression
Reza Ghane, Danil Akhtiamov, Babak Hassibi
https://arxiv.org/abs/2402.10474

One-Bit Quantization and Sparsification for Multiclass Linear Classification via Regularized Regression
We study the use of linear regression for multiclass classification in the over-parametrized regime where some of the training data is mislabeled. In such scenarios it is necessary to add an explicit regularization term, $λf(w)$, for some convex function $f(\cdot)$, to avoid overfitting the mislabeled data. In our analysis, we assume that the data is sampled from a Gaussian Mixture Model with equal class sizes, and that a proportion $c$ of the training labels is corrupted for each class. Under…

This https://arxiv.org/abs/2307.02551 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qu…

Semidefinite programming relaxations for quantum correlations
Semidefinite programs are convex optimisation problems involving a linear objective function and a domain of positive semidefinite matrices. Over the last two decades, they have become an indispensable tool in quantum information science. Many otherwise intractable fundamental and applied problems can be successfully approached by means of relaxation to a semidefinite program. Here, we review such methodology in the context of quantum correlations. We discuss how the core idea of semidefinite r…

This https://arxiv.org/abs/2207.02894 has been replaced.
link: https://scholar.google.com/scholar?q=a

Learning Acceptability Criteria from Good and Bad Decisions: An Inverse Optimization Approach
Conventional inverse optimization inputs a solution and finds the parameters of an optimization model that render a given solution optimal. The literature mostly focuses on inferring the objective function in linear problems when acceptable solutions are provided as input. In this paper, we propose an inverse optimization model that inputs several accepted and rejected solutions and recovers the underlying convex optimization model that can be used to generate such solutions. The novelty of our…

Non-asymptotic Global Convergence Analysis of BFGS with the Armijo-Wolfe Line Search
Qiujiang Jin, Ruichen Jiang, Aryan Mokhtari
https://arxiv.org/abs/2404.16731

Non-asymptotic Global Convergence Analysis of BFGS with the Armijo-Wolfe Line Search
In this paper, we establish the first explicit and non-asymptotic global convergence analysis of the BFGS method when deployed with an inexact line search scheme that satisfies the Armijo-Wolfe conditions. We show that BFGS achieves a global convergence rate of $(1-\frac{1}κ)^k$ for $μ$-strongly convex functions with $L$-Lipschitz gradients, where $κ=\frac{L}μ$ denotes the condition number. Furthermore, if the objective function's Hessian is Lipschitz, BFGS with the Armijo-Wolfe line search…

Equicontractive weak separation property on the line does not imply convex finite type condition
Kevin G. Hare
https://arxiv.org/abs/2403.00693 https://

Equicontractive weak separation property on the line does not imply convex finite type condition
Let $\{S_1, S_2, \dots, S_n\}$ be an iterated function system on $\mathbb{R}$ with attractor $K$. It is known that if the iterated function system satisfies the weak separation property and $K = [0,1]$ then the iterated function system also satisfies the convex finite type condition. We show that the condition $K = [0,1]$ is necessary. That is, we give two examples of iterated function systems on $\mathbb{R}$ satisfying weak separation condition, and $0< \dim_H(K) < 1$ such that the IFS does no…

Santal\'o Geometry of Convex Polytopes
Dmitrii Pavlov, Simon Telen
https://arxiv.org/abs/2402.18955 https://arxiv.org/pdf/2402.18…

Santaló Geometry of Convex Polytopes
The Santaló point of a convex polytope is the interior point which leads to a polar dual of minimal volume. This minimization problem is relevant in interior point methods for convex optimization, where the logarithm of the dual volume is known as the universal barrier function. When translating the facet hyperplanes, the Santaló point traces out a semi-algebraic set. We describe and compute this geometry using algebraic and numerical techniques. We exploit connections with statistics, optimi…

Overfitting Reduction in Convex Regression
Zhiqiang Liao, Sheng Dai, Eunji Lim, Timo Kuosmanen
https://arxiv.org/abs/2404.09528 https://

Overfitting Reduction in Convex Regression
Convex regression is a method for estimating an unknown function $f_0$ from a data set of $n$ noisy observations when $f_0$ is known to be convex. This method has played an important role in operations research, economics, machine learning, and many other areas. It has been empirically observed that the convex regression estimator produces inconsistent estimates of $f_0$ and extremely large subgradients near the boundary of the domain of $f_0$ as $n$ increases. In this paper, we provide theoret…

This https://arxiv.org/abs/2312.17628 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Joint User Scheduling, Power Allocation, and Rate Control for MC-RSMA in URLLC Services
This paper investigates the resource management problem in multi-carrier rate-splitting multiple access (MC-RSMA) systems with imperfect channel state information (CSI) and successive interference cancellation (SIC) for ultra-reliable and low-latency communications (URLLC) applications. To explore the trade-off between the decoding error probability and achievable rate, effective throughput (ET) is adopted as the utility function in this study. Then, a mixed-integer non-convex problem is formul…

This https://arxiv.org/abs/2310.06689 has been replaced.
link: https://scholar.google.com/scholar?q=a

Approximating Nash Equilibria in Normal-Form Games via Stochastic Optimization
We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.

This https://arxiv.org/abs/2207.12023 has been replaced.
link: https://scholar.google.com/scholar?q=a

A fast continuous time approach for non-smooth convex optimization with time scaling and Tikhonov regularization
In a Hilbert setting we aim to study a second order in time differential equation, combining viscous and Hessian-driven damping, containing a time scaling parameter function and a Tikhonov regularization term. The dynamical system is related to the problem of minimization of a nonsmooth convex function. In the formulation of the problem as well as in our analysis we use the Moreau envelope of the objective function and its gradient and heavily rely on their properties. We show that there is a s…

The limit as $s\nearrow 1$ of the fractional convex envelope
Bego\~na Barrios, Leandro M. Del Pezzo, Aleaxander Quaas, Julio D. Rossi
https://arxiv.org/abs/2404.07756

The limit as $s\nearrow 1$ of the fractional convex envelope
We study the behavior of the fractional convexity when the fractional parameter goes to 1. For any notion of convexity, the convex envelope of a datum prescribed on the boundary of a domain is defined as the largest possible convex function inside the domain that is below the datum on the boundary. Here we prove that the fractional convex envelope inside a strictly convex domain of a continuous and bounded exterior datum converges when $s\nearrow 1$ to the classical convex envelope of the restr…

Generalized Converses of Operator Jensens Inequalities with Applications to Hypercomplex Function Approximations and Bounds Algebra
Shih-Yu Chang
https://arxiv.org/abs/2404.11880 …

Generalized Converses of Operator Jensens Inequalities with Applications to Hypercomplex Function Approximations and Bounds Algebra
Mond and Pecaric proposed a powerful method, namd as MP method, to deal with operator inequalities. However, this method requires a real-valued function to be convex or concave, and the normalized positive linear map between Hilbert spaces. The objective of this study is to extend the MP method by allowing non-convex or non-concave real-valued functions and nonlinear mapping between Hilbert spaces. The Stone-Weierstrass theorem and Kantorovich function are fundamental components employed in gen…

Convexity properties related to Gauss hypergeometric function
Mohamed Bouali
https://arxiv.org/abs/2403.09695 https://arxiv.org/pdf/2…

Convexity properties related to Gauss hypergeometric function
We investigate the convexity property on $(0,1)$ of the functions $φ_{a,b,c}$ and $1/φ_{a,b,c}$, where $$φ_{a,b,c}(x)= \frac{c-\log(1-x)}{\,_2F_1(a,b,a+b,x)},$$ whenever $a,b\geq 0$ and $a+b\leq 1$. We Show that $φ_{a,b,c}$ (respectively $1/φ_{a,b,c}$) is strictly convex on $(0,1)$ if and only if $c\leq -2γ-ψ(a)-ψ(b),$ (respectively $c\geqα_0$) and $φ_{a,b,c}$ (respectively $1/φ_{a,b,c}$) is strictly concave on $(0,1)$ if and only if $c\geq c(a,b)$ (respectively $c\in[δ_-,δ_+]$), w…

Generalized Choi-Davis-Jensen's Operator Inequalities and Their Applications
Shih Yu Chang, Yimin Wei
https://arxiv.org/abs/2403.04892 https://<…

Generalized Choi-Davis-Jensen's Operator Inequalities and Their Applications
The original Choi-Davis-Jensen's inequality, with its wide-ranging applications in diverse scientific and engineering fields, has motivated researchers to explore generalizations. In this study, we extend Davis-Choi-Jensen's inequality by considering a nonlinear map instead of a normalized linear map and generalize operator convex function to any continuous function defined in a compact region. The Stone-Weierstrass theorem and Kantorovich function are instrumental in formulating and proving ge…

This https://arxiv.org/abs/2305.10620 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Soft-Minimum and Soft-Maximum Barrier Functions for Safety with Actuation Constraints
This paper presents two new control approaches for guaranteed safety (remaining in a safe set) subject to actuator constraints (the control is in a convex polytope). The control signals are computed using real-time optimization, including linear and quadratic programs subject to affine constraints, which are shown to be feasible. The first control method relies on a soft-minimum barrier function that is constructed using a finite-time-horizon prediction of the system trajectories under a known …

This https://arxiv.org/abs/2207.12023 has been replaced.
link: https://scholar.google.com/scholar?q=a

A fast continuous time approach for non-smooth convex optimization with time scaling and Tikhonov regularization
In a Hilbert setting we aim to study a second order in time differential equation, combining viscous and Hessian-driven damping, containing a time scaling parameter function and a Tikhonov regularization term. The dynamical system is related to the problem of minimization of a nonsmooth convex function. In the formulation of the problem as well as in our analysis we use the Moreau envelope of the objective function and its gradient and heavily rely on their properties. We show that there is a s…

This https://arxiv.org/abs/2310.15356 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Lie Group Variational Collision Integrators for a Class of Hybrid Systems
The problem of 3-dimensional, convex rigid-body collision over a plane is fully investigated; this includes bodies with sharp corners that is resolved without the need for nonsmooth convex analysis of tangent and normal cones. In particular, using nonsmooth Lagrangian mechanics, the equations of motion and jump equations are derived, which are largely dependent on the collision detection function. Following the variational approach, a Lie group variational collision integrator (LGVCI) is system…

This https://arxiv.org/abs/2304.04856 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Graph Convex Hull Bounds as generalized Jensen Inequalities
Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function $f\colon K \to \mathbb{R}$ defined on a convex domain $K \subseteq \mathbb{R}^{d}$ and any random variable $X$ taking values in $K$, $\mathbb{E}[f(X)] \geq f(\mathbb{E}[X])$. In this paper, sharp upper and lower bounds on $\mathbb{E}[f(X)]$, termed ``graph convex hull bounds'', are deri…

This https://arxiv.org/abs/2310.06689 has been replaced.
link: https://scholar.google.com/scholar?q=a

Approximating Nash Equilibria in Normal-Form Games via Stochastic Optimization
We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.

Localization in boundary-driven lattice models
Michele Giusfredi, Stefano Iubini, Paolo Politi
https://arxiv.org/abs/2404.12159 https://

Localization in boundary-driven lattice models
Several systems may display an equilibrium condensation transition, where a finite fraction of a conserved quantity is spatially localized. The presence of two conservation laws may induce the emergence of such transition in an out-of-equilibrium setup, where boundaries are attached to two different and subcritical heat baths. We study this phenomenon in a class of stochastic lattice models, where the local energy is a general convex function of the local mass, mass and energy being both global…

This https://arxiv.org/abs/2401.11245 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Construction of the log-convex minorant of a sequence $\{M_α\}_{α\in\mathbb{N}_0^d}$
We give a simple construction of the log-convex minorant of a sequence $\{M_α\}_{α\in\mathbb{N}_0^d}$ and consequently extend to the $d$-dimensional case the well-known formula that relates a log-convex sequence $\{M_p\}_{p\in\mathbb{N}_0}$ to its associated function $ω_M$, that is $M_p=\sup_{t>0}t^p\exp(-ω_M(t))$. We show that in the more dimensional anisotropic case the classical log-convex condition $M_α^2\leq M_{α-e_j}M_{α+e_j}$ is not sufficient: convexity as a function of more vari…

This https://arxiv.org/abs/2308.04359 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A characterization of bounded balanced convex domains in $\mathbb{C}^n$
In this paper, we investigate the characterization of balanced bounded convex domains in $\mathbb{C}^n$ in terms of the squeezing function. As an application, we provide a characterization of the polydisc in $\mathbb{C}^n$.

Single-loop Projection-free and Projected Gradient-based Algorithms for Nonconvex-concave Saddle Point Problems with Bilevel Structure
Mohammad Mahdi Ahmadi, Erfan Yazdandoost Hamedani
https://arxiv.org/abs/2404.13021

Single-loop Projection-free and Projected Gradient-based Algorithms for Nonconvex-concave Saddle Point Problems with Bilevel Structure
In this paper, we explore a class of constrained saddle point problems with a bilevel structure, where the upper-level objective function is nonconvex-concave and smooth subject to a strongly convex lower-level objective function. This class of problems finds wide applicability in machine learning, including robust multi-task learning, adversarial learning, and robust meta-learning. While some studies have focused on simpler formulations, e.g., when the upper-level objective function is linear …

This https://arxiv.org/abs/2402.19212 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Deep Reinforcement Learning: A Convex Optimization Approach
In this paper, we consider reinforcement learning of nonlinear systems with continuous state and action spaces. We present an episodic learning algorithm, where we for each episode use convex optimization to find a two-layer neural network approximation of the optimal $Q$-function. The convex optimization approach guarantees that the weights calculated at each episode are optimal, with respect to the given sampled states and actions of the current episode. For stable nonlinear systems, we show …

Localization in boundary-driven lattice models
Michele Giusfredi, Stefano Iubini, Paolo Politi
https://arxiv.org/abs/2404.12159 https://

Localization in boundary-driven lattice models
Several systems may display an equilibrium condensation transition, where a finite fraction of a conserved quantity is spatially localized. The presence of two conservation laws may induce the emergence of such transition in an out-of-equilibrium setup, where boundaries are attached to two different and subcritical heat baths. We study this phenomenon in a class of stochastic lattice models, where the local energy is a general convex function of the local mass, mass and energy being both global…

This https://arxiv.org/abs/2305.12643 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_sta…

A two-way heterogeneity model for dynamic networks
Dynamic network data analysis requires joint modelling individual snapshots and time dynamics. This paper proposes a new two-way heterogeneity model towards this goal. The new model equips each node of the network with two heterogeneity parameters, one to characterize the propensity of forming ties with other nodes and the other to differentiate the tendency of retaining existing ties over time. Though the negative log-likelihood function is non-convex, it is locally convex in a neighbourhood o…

This https://arxiv.org/abs/2201.02323 has been replaced.
link: https://scholar.google.com/scholar?q=a

Distributed Nash Equilibrium Seeking over Time-Varying Directed Communication Networks
We study distributed algorithms for finding a Nash equilibrium (NE) in a class of non-cooperative convex games under partial information. Specifically, each agent has access only to its own smooth local cost function and can receive information from its neighbors in a time-varying directed communication network. To this end, we propose a distributed gradient play algorithm to compute a NE by utilizing local information exchange among the players. In this algorithm, every agent performs a gradie…

This https://arxiv.org/abs/2308.01753 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Minimal Convex Environmental Contours
We develop a numerical method for the computation of a minimal convex and compact set, $\mathcal{B}\subset\mathbb{R}^N$, in the sense of mean width. This minimisation is constrained by the requirement that $\max_{b\in\mathcal{B}}\langle b , u\rangle\geq C(u)$ for all unit vectors $u\in S^{N-1}$ given some Lipschitz function $C$. This problem arises in the construction of environmental contours under the assumption of convex failure sets. Environmental contours offer descriptions of extreme en…

Global Complexity Analysis of BFGS
Anton Rodomanov
https://arxiv.org/abs/2404.15051 https://arxiv.org/pdf/2404.15051

Global Complexity Analysis of BFGS
In this paper, we present a global complexity analysis of the classical BFGS method with inexact line search, as applied to minimizing a strongly convex function with Lipschitz continuous gradient and Hessian. We consider a variety of standard line search strategies including the backtracking line search based on the Armijo condition, Armijo-Goldstein and Wolfe-Powell line searches. Our analysis suggests that the convergence of the algorithm proceeds in several different stages before the fast …

This https://arxiv.org/abs/2401.07968 has been replaced.
link: https://scholar.google.com/scholar?q=a

Characterizing the minimax rate of nonparametric regression under bounded convex constraints
We quantify the minimax rate for a nonparametric regression model over a convex function class $\mathcal{F}$ with bounded diameter. We obtain a minimax rate of ${\varepsilon^{\ast}}^2\wedge\mathrm{diam}(\mathcal{F})^2$ where \[\varepsilon^{\ast} =\sup\{\varepsilon>0:n\varepsilon^2 \le \log M_{\mathcal{F}}^{\operatorname{loc}}(\varepsilon,c)\},\] where $M_{\mathcal{F}}^{\operatorname{loc}}(\cdot, c)$ is the local metric entropy of $\mathcal{F}$ and our loss function is the squared population $L_…

This https://arxiv.org/abs/2309.09505 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Outlier-Insensitive Kalman Filtering: Theory and Applications
State estimation of dynamical systems from noisy observations is a fundamental task in many applications. It is commonly addressed using the linear Kalman filter (KF), whose performance can significantly degrade in the presence of outliers in the observations, due to the sensitivity of its convex quadratic objective function. To mitigate such behavior, outlier detection algorithms can be applied. In this work, we propose a parameter-free algorithm which mitigates the harmful effect of outliers …

This https://arxiv.org/abs/2308.02070 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Nonlinearly Elastic Maps: Energy Minimizing Configurations of Membranes on Prescribed Surfaces
We propose a model for nonlinearly elastic membranes undergoing finite deformations while confined to a regular frictionless surface in $\mathbb{R}^3$. This is a physically correct model of the analogy sometimes given to motivate harmonic maps between manifolds. The proposed stored-energy function is convex in the strain pair comprising the deformation gradient and the local area ratio. If the target surface is a plane, the problem reduces to 2-dimensional, polyconvex nonlinear elasticity addre…

A Relative Inexact Proximal Gradient Method With an Explicit Linesearch
Yunier Bello-Cruz, Max L. N. Gon\c{c}alves, Jefferson G. Melo, Cassandra Mohr
https://arxiv.org/abs/2404.10987

A Relative Inexact Proximal Gradient Method With an Explicit Linesearch
This paper presents and investigates an inexact proximal gradient method for solving composite convex optimization problems characterized by an objective function composed of a sum of a full domain differentiable convex function and a non-differentiable convex function. We introduce an explicit linesearch strategy that requires only a relative inexact solution of the proximal subproblem per iteration. We prove the convergence of the sequence generated by our scheme and establish its iteration c…

This https://arxiv.org/abs/2305.12643 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_sta…

A two-way heterogeneity model for dynamic networks
Dynamic network data analysis requires joint modelling individual snapshots and time dynamics. This paper proposes a new two-way heterogeneity model towards this goal. The new model equips each node of the network with two heterogeneity parameters, one to characterize the propensity of forming ties with other nodes and the other to differentiate the tendency of retaining existing ties over time. Though the negative log-likelihood function is non-convex, it is locally convex in a neighbourhood o…

This https://arxiv.org/abs/2201.02323 has been replaced.
link: https://scholar.google.com/scholar?q=a

Distributed Nash Equilibrium Seeking over Time-Varying Directed Communication Networks
We study distributed algorithms for finding a Nash equilibrium (NE) in a class of non-cooperative convex games under partial information. Specifically, each agent has access only to its own smooth local cost function and can receive information from its neighbors in a time-varying directed communication network. To this end, we propose a distributed gradient play algorithm to compute a NE by utilizing local information exchange among the players. In this algorithm, every agent performs a gradie…

This https://arxiv.org/abs/2309.09505 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Outlier-Insensitive Kalman Filtering: Theory and Applications
State estimation of dynamical systems from noisy observations is a fundamental task in many applications. It is commonly addressed using the linear Kalman filter (KF), whose performance can significantly degrade in the presence of outliers in the observations, due to the sensitivity of its convex quadratic objective function. To mitigate such behavior, outlier detection algorithms can be applied. In this work, we propose a parameter-free algorithm which mitigates the harmful effect of outliers …

This https://arxiv.org/abs/1905.04480 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Real-Valued Bochner integral and the Modern Real-Valued Measurable function on $\mathbb{R}$
The like-Lebesgue integral of real-valued measurable functions (abbreviated as \textit{RVM-MI})is the most complete and appropriate integration Theory. Integrals are also defined in abstract spaces since Pettis (1938). In particular, Bochner integrals received much interest with very recent researches. It is very commode to use the \textit{RVM-MI} in constructing Bochner integral in Banach or in locally convex spaces. In this simple not, we prove that the Bochner integral and the \textit{RVM-MI…

Global optimality under amenable symmetry constraints
Peter Orbanz
https://arxiv.org/abs/2402.07613 https://arxiv.org/pdf/2402.07613<…

Global optimality under amenable symmetry constraints
We ask whether there exists a function or measure that (1) minimizes a given convex functional or risk and (2) satisfies a symmetry property specified by an amenable group of transformations. Examples of such symmetry properties are invariance, equivariance, or quasi-invariance. Our results draw on old ideas of Stein and Le Cam and on approximate group averages that appear in ergodic theorems for amenable groups. A class of convex sets known as orbitopes in convex analysis emerges as crucial, a…

This https://arxiv.org/abs/2311.02671 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Extension of convex functions from a hyperplane to a half-space
It is shown that a possibly infinite-valued proper lower semicontinuous convex function on ${\mathbb R}^n$ has an extension to a convex function on the half-space ${\mathbb R}^n\times[0,\infty)$ which is finite and smooth on the open half-space ${\mathbb R}^n\times(0,\infty)$. The result is applied to nonlinear elasticity, where it clarifies how the condition of polyconvexity of the free-energy density $ψ(Dy)$ is best expressed when $ψ(A)\to\infty$ as $\det A\to 0+$.

This https://arxiv.org/abs/2205.13055 has been replaced.
link: https://scholar.google.com/scholar?q=a

Complexity-optimal and Parameter-free First-order Methods for Finding Stationary Points of Composite Optimization Problems
This paper develops and analyzes an accelerated proximal descent method for finding stationary points of nonconvex composite optimization problems. The objective function is of the form $f+h$ where $h$ is a proper closed convex function, $f$ is a differentiable function on the domain of $h$, and $\nabla f$ is Lipschitz continuous on the domain of $h$. The main advantage of this method is that it is "parameter-free" in the sense that it does not require knowledge of the Lipschitz constant of $\n…

This https://arxiv.org/abs/2210.09665 has been replaced.
link: https://scholar.google.com/scholar?q=a

On convergence of a $q$-random coordinate constrained algorithm for non-convex problems
We propose a random coordinate descent algorithm for optimizing a non-convex objective function subject to one linear constraint and simple bounds on the variables. Although it is common use to update only two random coordinates simultaneously in each iteration of a coordinate descent algorithm, our algorithm allows updating arbitrary number of coordinates. We provide a proof of convergence of the algorithm. The convergence rate of the algorithm improves when we update more coordinates per iter…

Global optimality under amenable symmetry constraints
Peter Orbanz
https://arxiv.org/abs/2402.07613 https://arxiv.org/pdf/2402.07613<…

Global optimality under amenable symmetry constraints
We ask whether there exists a function or measure that (1) minimizes a given convex functional or risk and (2) satisfies a symmetry property specified by an amenable group of transformations. Examples of such symmetry properties are invariance, equivariance, or quasi-invariance. Our results draw on old ideas of Stein and Le Cam and on approximate group averages that appear in ergodic theorems for amenable groups. A class of convex sets known as orbitopes in convex analysis emerges as crucial, a…

Winner-Pays-Bid Auctions Minimize Variance
Preston McAfee, Renato Paes Leme, Balasubramanian Sivan, Sergei Vassilvitskii
https://arxiv.org/abs/2403.04856 h…

Winner-Pays-Bid Auctions Minimize Variance
Any social choice function (e.g the efficient allocation) can be implemented using different payment rules: first price, second price, all-pay, etc. All of these payment rules are guaranteed to have the same expected revenue by the revenue equivalence theorem, but have different distributions of revenue, leading to a question of which one is best. We prove that among all possible payment rules, winner-pays-bid minimizes the variance in revenue and, in fact, minimizes any convex risk measure.

This https://arxiv.org/abs/2403.09044 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Relationship between General MP and DPP for the Stochastic Recursive Optimal Control Problem With Jumps: Viscosity Solution Framework
This paper is concerned with the relationship between general maximum principle and dynamic programming principle for the stochastic recursive optimal control problem with jumps, where the control domain is not necessarily convex. Relations among the adjoint processes, the generalized Hamiltonian function and the value function are proved, under the assumption of a smooth value function and within the framework of viscosity solutions, respectively. Some examples are given to illustrate the theo…

Tikhonov Regularization for Stochastic Non-Smooth Convex Optimization in Hilbert Spaces
Rodrigo Maulen-Soto, Jalal Fadili, Hedy Attouch
https://arxiv.org/abs/2403.06708

Tikhonov Regularization for Stochastic Non-Smooth Convex Optimization in Hilbert Spaces
To solve non-smooth convex optimization problems with a noisy gradient input, we analyze the global behavior of subgradient-like flows under stochastic errors. The objective function is composite, being equal to the sum of two convex functions, one being differentiable and the other potentially non-smooth. We then use stochastic differential inclusions where the drift term is minus the subgradient of the objective function, and the diffusion term is either bounded or square-integrable. In this …

Revisiting Convergence of AdaGrad with Relaxed Assumptions
Yusu Hong, Junhong Lin
https://arxiv.org/abs/2402.13794 https://arxiv.org/…

Revisiting Convergence of AdaGrad with Relaxed Assumptions
In this study, we revisit the convergence of AdaGrad with momentum (covering AdaGrad as a special case) on non-convex smooth optimization problems. We consider a general noise model where the noise magnitude is controlled by the function value gap together with the gradient magnitude. This model encompasses a broad range of noises including bounded noise, sub-Gaussian noise, affine variance noise and the expected smoothness, and it has been shown to be more realistic in many practical applicati…

Heavy Ball Momentum for Non-Strongly Convex Optimization
Jean-Fran\c{c}ois Aujol, Charles Dossal, Hippolyte Labarri\`ere, Aude Rondepierre
https://arxiv.org/abs/2403.06930

Heavy Ball Momentum for Non-Strongly Convex Optimization
When considering the minimization of a quadratic or strongly convex function, it is well known that first-order methods involving an inertial term weighted by a constant-in-time parameter are particularly efficient (see Polyak [32], Nesterov [28], and references therein). By setting the inertial parameter according to the condition number of the objective function, these methods guarantee a fast exponential decay of the error. We prove that this type of schemes (which are later called Heavy Bal…

On the convergence of Block Majorization-Minimization algorithms on the Grassmann Manifold
Carlos Alejandro Lopez, Jaume Riba
https://arxiv.org/abs/2402.09907

On the convergence of Block Majorization-Minimization algorithms on the Grassmann Manifold
The Majorization-Minimization (MM) framework is widely used to derive efficient algorithms for specific problems that require the optimization of a cost function (which can be convex or not). It is based on a sequential optimization of a surrogate function over closed convex sets. A natural extension of this framework incorporates ideas of Block Coordinate Descent (BCD) algorithms into the MM framework, also known as block MM. The rationale behind the block extension is to partition the optimiz…

A Proximal Gradient Method with an Explicit Line search for Multiobjective Optimization
Yunier Bello-Cruz, J. G. Melo, L. F. Prudente, R. V. G. Serra
https://arxiv.org/abs/2404.10993

A Proximal Gradient Method with an Explicit Line search for Multiobjective Optimization
We present a proximal gradient method for solving convex multiobjective optimization problems, where each objective function is the sum of two convex functions, with one assumed to be continuously differentiable. The algorithm incorporates a backtracking line search procedure that requires solving only one proximal subproblem per iteration, and is exclusively applied to the differentiable part of the objective functions. Under mild assumptions, we show that the sequence generated by the method …

A New Algorithm With Lower Complexity for Bilevel Optimization
Haimei Huo, Zhixun Su
https://arxiv.org/abs/2404.11377 https://arxiv.o…

A New Algorithm With Lower Complexity for Bilevel Optimization
Many stochastic algorithms have been proposed to solve the bilevel optimization problem, where the lower level function is strongly convex and the upper level value function is nonconvex. In particular, exising Hessian inverse-free algorithms that utilize momentum recursion or variance reduction technqiues can reach an $ε$-stationary point with a complexity of $\tilde{O}(ε^{-1.5})$ under usual smoothness conditions. However, $\tilde{O}(ε^{-1.5})$ is a complexity higher than $O(ε^{-1.5})$. H…

This https://arxiv.org/abs/2302.13970 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications
We study the problem of estimating the convex hull of the image $f(X)\subset\mathbb{R}^n$ of a compact set $X\subset\mathbb{R}^m$ with smooth boundary through a smooth function $f:\mathbb{R}^m\to\mathbb{R}^n$. Assuming that $f$ is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of $f(X)$ and the convex hull of the images $f(x_i)$ of $M$ sampled inputs $x_i$ on the boundary of $X$. When applied to the problem of geometric inference from a random sample, our …

Relationship between General MP and DPP for the Stochastic Recursive Optimal Control Problem With Jumps: Viscosity Solution Framework
Bin Wang, Jiingtao Shi
https://arxiv.org/abs/2403.09044

Relationship between General MP and DPP for the Stochastic Recursive Optimal Control Problem With Jumps: Viscosity Solution Framework
This paper is concerned with the relationship between general maximum principle and dynamic programming principle for the stochastic recursive optimal control problem with jumps, where the control domain is not necessarily convex. Relations among the adjoint processes, the generalized Hamiltonian function and the value function are proved, under the assumption of a smooth value function and within the framework of viscosity solutions, respectively. Some examples are given to illustrate the theo…

On Weakly Contracting Dynamics for Convex Optimization
Veronica Centorrino, Alexander Davydov, Anand Gokhale, Giovanni Russo, Francesco Bullo
https://arxiv.org/abs/2403.07572

On Weakly Contracting Dynamics for Convex Optimization
We investigate the convergence characteristics of dynamics that are \emph{globally weakly} and \emph{locally strongly contracting}. Such dynamics naturally arise in the context of convex optimization problems with a unique minimizer. We show that convergence to the equilibrium is \emph{linear-exponential}, in the sense that the distance between each solution and the equilibrium is upper bounded by a function that first decreases linearly and then exponentially. As we show, the linear-exponentia…

On Weakly Contracting Dynamics for Convex Optimization
Veronica Centorrino, Alexander Davydov, Anand Gokhale, Giovanni Russo, Francesco Bullo
https://arxiv.org/abs/2403.07572

On Weakly Contracting Dynamics for Convex Optimization
We investigate the convergence characteristics of dynamics that are \emph{globally weakly} and \emph{locally strongly contracting}. Such dynamics naturally arise in the context of convex optimization problems with a unique minimizer. We show that convergence to the equilibrium is \emph{linear-exponential}, in the sense that the distance between each solution and the equilibrium is upper bounded by a function that first decreases linearly and then exponentially. As we show, the linear-exponentia…

This https://arxiv.org/abs/2311.10053 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Near-optimal Closed-loop Method via Lyapunov Damping for Convex Optimization
We introduce an autonomous system with closed-loop damping for first-order convex optimization. While, to this day, optimal rates of convergence are almost exclusively achieved by non-autonomous methods via open-loop damping (e.g., Nesterov's algorithm), we show that our system, featuring a closed-loop damping, exhibits a rate arbitrarily close to the optimal one. We do so by coupling the damping and the speed of convergence of the system via a well-chosen Lyapunov function. By discretizing our…

This https://arxiv.org/abs/2311.10053 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Near-optimal Closed-loop Method via Lyapunov Damping for Convex Optimization
We introduce an autonomous system with closed-loop damping for first-order convex optimization. While, to this day, optimal rates of convergence are almost exclusively achieved by non-autonomous methods via open-loop damping (e.g., Nesterov's algorithm), we show that our system, featuring a closed-loop damping, exhibits a rate arbitrarily close to the optimal one. We do so by coupling the damping and the speed of convergence of the system via a well-chosen Lyapunov function. By discretizing our…

On the Set of Possible Minimizers of a Sum of Convex Functions
Moslem Zamani, Fran\c{c}ois Glineur, Julien M. Hendrickx
https://arxiv.org/abs/2403.05467 ht…

On the Set of Possible Minimizers of a Sum of Convex Functions
Consider a sum of convex functions, where the only information known about each individual summand is the location of a minimizer. In this work, we give an exact characterization of the set of possible minimizers of the sum. Our results cover several types of assumptions on the summands, such as smoothness or strong convexity. Our main tool is the use of necessary and sufficient conditions for interpolating the considered function classes, which leads to shorter and more direct proofs in compar…

Gaining or losing perspective for convex multivariate functions on box domains
Luze Xu, Jon Lee
https://arxiv.org/abs/2404.07010 https://

Gaining or losing perspective for convex multivariate functions on box domains
MINLO (mixed-integer nonlinear optimization) formulations of the disjunction between the origin and a polytope via a binary indicator variable is broadly used in nonlinear combinatorial optimization for modeling a fixed cost associated with carrying out a group of activities and a convex cost function associated with the levels of the activities. The perspective relaxation of such models is often used to solve to global optimality in a branch-and-bound context, but it typically requires suitabl…

Low-Rank Extragradient Methods for Scalable Semidefinite Optimization
Dan Garber. Atara Kaplan
https://arxiv.org/abs/2402.09081 https://

Low-Rank Extragradient Methods for Scalable Semidefinite Optimization
We consider several classes of highly important semidefinite optimization problems that involve both a convex objective function (smooth or nonsmooth) and additional linear or nonlinear smooth and convex constraints, which are ubiquitous in statistics, machine learning, combinatorial optimization, and other domains. We focus on high-dimensional and plausible settings in which the problem admits a low-rank solution which also satisfies a low-rank complementarity condition. We provide several the…

This https://arxiv.org/abs/2402.00166 has been replaced.
link: https://scholar.google.com/scholar?q=a

Network Design for the Traffic Assignment Problem with Mixed-Integer Frank-Wolfe
We tackle the network design problem for centralized traffic assignment, which can be cast as a mixed-integer convex optimization (MICO) problem. For this task, we propose different formulations and solution methods in both a deterministic and a stochastic setting in which the demand is unknown in the design phase. We leverage the recently proposed Boscia framework, which can solve MICO problems when the main nonlinearity stems from a differentiable objective function. Boscia tackles these prob…

This https://arxiv.org/abs/2208.13190 has been replaced.
link: https://scholar.google.com/scholar?q=a

Exploiting higher-order derivatives in convex optimization methods
Exploiting higher-order derivatives in convex optimization is known at least since 1970's. In each iteration higher-order (also called tensor) methods minimize a regularized Taylor expansion of the objective function, which leads to faster convergence rates if the corresponding higher-order derivative is Lipschitz-continuous. Recently a series of lower iteration complexity bounds for such methods were proved, and a gap between upper an lower complexity bounds was revealed. Moreover, it was show…

This https://arxiv.org/abs/2308.13776 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A VMiPG method for composite optimization with nonsmooth term having no closed-form proximal mapping
This paper concerns the minimization of the sum of a twice continuously differentiable function $f$ and a nonsmooth convex function $g$ without closed-form proximal mapping. For this class of nonconvex and nonsmooth problems, we propose a line-search based variable metric inexact proximal gradient (VMiPG) method with uniformly bounded positive definite variable metric linear operators. This method computes in each step an inexact minimizer of a strongly convex model such that the difference bet…

This https://arxiv.org/abs/2402.19212 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Deep Reinforcement Learning: A Convex Optimization Approach
In this paper, we consider reinforcement learning of nonlinear systems with continuous state and action spaces. We present an episodic learning algorithm, where we for each episode use convex optimization to find a two-layer neural network approximation of the optimal $Q$-function. The convex optimization approach guarantees that the weights calculated at each episode are optimal, with respect to the given sampled states and actions of the current episode. For stable nonlinear systems, we show …

This https://arxiv.org/abs/2402.05415 has been replaced.
link: https://scholar.google.com/scholar?q=a

Near-Optimal Convex Simple Bilevel Optimization with a Bisection Method
This paper studies a class of simple bilevel optimization problems where we minimize a composite convex function at the upper-level subject to a composite convex lower-level problem. Existing methods either provide asymptotic guarantees for the upper-level objective or attain slow sublinear convergence rates. We propose a bisection algorithm to find a solution that is $ε_f$-optimal for the upper-level objective and $ε_g$-optimal for the lower-level objective. In each iteration, the binary sea…

Deep Reinforcement Learning: A Convex Optimization Approach
Ather Gattami
https://arxiv.org/abs/2402.19212 https://arxiv.org/pdf/2402…

Deep Reinforcement Learning: A Convex Optimization Approach
In this paper, we consider reinforcement learning of nonlinear systems with continuous state and action spaces. We present an episodic learning algorithm, where we for each episode use convex optimization to find a two-layer neural network approximation of the optimal $Q$-function. The convex optimization approach guarantees that the weights calculated at each episode are optimal, with respect to the given sampled states and actions of the current episode. For stable nonlinear systems, we show …

This https://arxiv.org/abs/2306.16255 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Theory and applications of the Sum-Of-Squares technique
The Sum-of-Squares (SOS) approximation method is a technique used in optimization problems to derive lower bounds on the optimal value of an objective function. By representing the objective function as a sum of squares in a feature space, the SOS method transforms non-convex global optimization problems into solvable semidefinite programs. This note presents an overview of the SOS method. We start with its application in finite-dimensional feature spaces and, subsequently, we extend it to infi…

Adaptive generalized conditional gradient method for multiobjective optimization
Anteneh Getachew Gebrie, Ellen Hidemi Fukuda
https://arxiv.org/abs/2404.04174

Adaptive generalized conditional gradient method for multiobjective optimization
In this paper, we propose a generalized conditional gradient method for multiobjective optimization, which can be viewed as an improved extension of the classical Frank-Wolfe (conditional gradient) method for single-objective optimization. The proposed method works for both constrained and unconstrained benchmark multiobjective optimization problems, where the objective function is the summation of a smooth function and a possibly nonsmooth convex function. The method combines the so-called nor…

Proximal Oracles for Optimization and Sampling
Jiaming Liang, Yongxin Chen
https://arxiv.org/abs/2404.02239 https://arxiv.org/pdf/240…

Proximal Oracles for Optimization and Sampling
We consider convex optimization with non-smooth objective function and log-concave sampling with non-smooth potential (negative log density). In particular, we study two specific settings where the convex objective/potential function is either semi-smooth or in composite form as the finite sum of semi-smooth components. To overcome the challenges caused by non-smoothness, our algorithms employ two powerful proximal frameworks in optimization and sampling: the proximal point framework for optimi…

This https://arxiv.org/abs/2307.04504 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

An Algorithm with Optimal Dimension-Dependence for Zero-Order Nonsmooth Nonconvex Stochastic Optimization
We study the complexity of producing $(δ,ε)$-stationary points of Lipschitz objectives which are possibly neither smooth nor convex, using only noisy function evaluations. Recent works proposed several stochastic zero-order algorithms that solve this task, all of which suffer from a dimension-dependence of $Ω(d^{3/2})$ where $d$ is the dimension of the problem, which was conjectured to be optimal. We refute this conjecture by providing a faster algorithm that has complexity $O(dδ^{-1}ε^{-3…

This https://arxiv.org/abs/2307.04504 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

An Algorithm with Optimal Dimension-Dependence for Zero-Order Nonsmooth Nonconvex Stochastic Optimization
We study the complexity of producing $(δ,ε)$-stationary points of Lipschitz objectives which are possibly neither smooth nor convex, using only noisy function evaluations. Recent works proposed several stochastic zero-order algorithms that solve this task, all of which suffer from a dimension-dependence of $Ω(d^{3/2})$ where $d$ is the dimension of the problem, which was conjectured to be optimal. We refute this conjecture by providing a faster algorithm that has complexity $O(dδ^{-1}ε^{-3…

The largest-$K$-norm for general measure spaces and a DC Reformulation for $L^0$-Constrained Problems in Function Spaces
Bastian Dittrich, Daniel Wachsmuth
https://arxiv.org/abs/2403.19437

The largest-$K$-norm for general measure spaces and a DC Reformulation for $L^0$-Constrained Problems in Function Spaces
We consider constraints on the measure of the support for integrable functions on arbitrary measure spaces. It is shown that this non-convex and discontinuous constraint can be equivalently reformulated by the difference of two convex and continuous functions, namely the $L^1$-norm and the so-called largest-$K$-norm. The largest-$K$-norm is studied and its convex subdifferential is derived. An exemplary problem is solved by applying a DC method to the reformulated problem.

This https://arxiv.org/abs/2403.06708 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Tikhonov Regularization for Stochastic Non-Smooth Convex Optimization in Hilbert Spaces
To solve non-smooth convex optimization problems with a noisy gradient input, we analyze the global behavior of subgradient-like flows under stochastic errors. The objective function is composite, being equal to the sum of two convex functions, one being differentiable and the other potentially non-smooth. We then use stochastic differential inclusions where the drift term is minus the subgradient of the objective function, and the diffusion term is either bounded or square-integrable. In this …

This https://arxiv.org/abs/2403.06708 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Tikhonov Regularization for Stochastic Non-Smooth Convex Optimization in Hilbert Spaces
To solve non-smooth convex optimization problems with a noisy gradient input, we analyze the global behavior of subgradient-like flows under stochastic errors. The objective function is composite, being equal to the sum of two convex functions, one being differentiable and the other potentially non-smooth. We then use stochastic differential inclusions where the drift term is minus the subgradient of the objective function, and the diffusion term is either bounded or square-integrable. In this …

This https://arxiv.org/abs/2404.02635 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

An Inexact Regularized Proximal Newton Method without Line Search
In this paper, we introduce an inexact regularized proximal Newton method (IRPNM) that does not require any line search. The method is designed to minimize the sum of a twice continuously differentiable function $f$ and a convex (possibly non-smooth and extended-valued) function $φ$. Instead of controlling a step size by a line search procedure, we update the regularization parameter in a suitable way, based on the success of the previous iteration. The global convergence of the sequence of it…

Polyhedral Analysis of Quadratic Optimization Problems with Stieltjes Matrices and Indicators
Peijing Liu, Alper Atamt\"urk, Andr\'es G\'omez, Simge K\"u\c{c}\"ukyavuz
https://arxiv.org/abs/2404.04236

Polyhedral Analysis of Quadratic Optimization Problems with Stieltjes Matrices and Indicators
In this paper, we consider convex quadratic optimization problems with indicators on the continuous variables. In particular, we assume that the Hessian of the quadratic term is a Stieltjes matrix, which naturally appears in sparse graphical inference problems and others. We describe an explicit convex formulation for the problem by studying the Stieltjes polyhedron arising as part of an extended formulation and exploiting the supermodularity of a set function defined on its extreme points. Our…

This https://arxiv.org/abs/2402.19212 has been replaced.
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Deep Reinforcement Learning: A Convex Optimization Approach
In this paper, we consider reinforcement learning of nonlinear systems with continuous state and action spaces. We present an episodic learning algorithm, where we for each episode use convex optimization to find a two-layer neural network approximation of the optimal $Q$-function. The convex optimization approach guarantees that the weights calculated at each episode are optimal, with respect to the given sampled states and actions of the current episode. For stable nonlinear systems, we show …