Tootfinder

Hardness and Algorithmic Results for Roman \{3\}-Domination
Sangam Balchandar Reddy
https://arxiv.org/abs/2509.23615 https://arxiv.org/pdf/2509.23615

Hardness and Algorithmic Results for Roman \{3\}-Domination
A Roman $\{3\}$-dominating function on a graph $G = (V, E)$ is a function $f: V \rightarrow \{0, 1, 2, 3\}$ such that for each vertex $u \in V$, if $f(u) = 0$ then $\sum_{v \in N(u)} f(v) \geq 3$ and if $f(u) = 1$ then $\sum_{v \in N(u)} f(v) \geq 2$. The weight of a Roman $\{3\}$-dominating function $f$ is $\sum_{u \in V} f(u)$. The objective of \rtd{} is to compute a Roman $\{3\}$-dominating function of minimum weight. The problem is known to be NP-complete on chordal graphs, star-convex bipa…

Generalization Analysis for Classification on Korobov Space
Yuqing Liu
https://arxiv.org/abs/2509.22748 https://arxiv.org/pdf/2509.22748

Generalization Analysis for Classification on Korobov Space
In this paper, the classification algorithm arising from Tikhonov regularization is discussed. The main intention is to derive learning rates for the excess misclassification error according to the convex $η$-norm loss function $ϕ(v)=(1 - v)_{+}^η$, $η\geq1$. Following the argument, the estimation of error under Tsybakov noise conditions is studied. In addition, we propose the rate of $L_p$ approximation of functions from Korobov space $X^{2, p}([-1,1]^{d})$, $1\leq p \leq \infty$, by the s…

Accelerated stochastic first-order method for convex optimization under heavy-tailed noise
Chuan He, Zhaosong Lu
https://arxiv.org/abs/2510.11676 https://a…

Accelerated stochastic first-order method for convex optimization under heavy-tailed noise
We study convex composite optimization problems, where the objective function is given by the sum of a prox-friendly function and a convex function whose subgradients are estimated under heavy-tailed noise. Existing work often employs gradient clipping or normalization techniques in stochastic first-order methods to address heavy-tailed noise. In this paper, we demonstrate that a vanilla stochastic algorithm -- without additional modifications such as clipping or normalization -- can achieve op…

Phase Transitions of the Additive Uniform Noise Channel with Peak Amplitude and Cost Constraint
Jonas Stapmanns, Catarina Dias, Luke Eilers, Tobias K\"uhn, Jean-Pascal Pfister
https://arxiv.org/abs/2510.12427

Phase Transitions of the Additive Uniform Noise Channel with Peak Amplitude and Cost Constraint
Under which condition is quantization optimal? We address this question in the context of the additive uniform noise channel under peak amplitude and cost constraints. We compute analytically the capacity-achieving input distribution as a function of the noise level, the average cost constraint, and the curvature of the cost function. We find that when the cost function is concave, the capacity-achieving input distribution is discrete, whereas when the cost function is convex and the cost const…

On Convex Functions of Gaussian Variables
Maite Fern\'andez-Unzueta, James Melbourne, Gerardo Palafox-Castillo
https://arxiv.org/abs/2510.06676 https://

On Convex Functions of Gaussian Variables
We investigate a convexity properties for normalized log moment generating function continuing a recent investigation of Chen of convex images of Gaussians. We show that any variable satisfying a ``Ehrhard-like'' property for its distribution function has a strictly convex normalized log moment generating function, unless the variable is Gaussian, in which case affine-ness is achieved. Moreover we characterize variables that satisfy the Ehrhard-like property as the convex images of Gaussians. A…

An Efficient Solution Method for Solving Convex Separable Quadratic Optimization Problems
Shaoze Li, Junhao Wu, Cheng Lu, Zhibin Deng, Shu-Cherng Fang
https://arxiv.org/abs/2510.11554

An Efficient Solution Method for Solving Convex Separable Quadratic Optimization Problems
Convex separable quadratic optimization problems occur in many practical applications. In this paper, based on an iterative resolution scheme of the KKT system, we develop an efficient method for solving a quadratic programming problem with a convex separable objective function subject to multiple convex separable constraints. We show that the proposed approach leads to a dual coordinate ascent algorithm and provide a convergence proof. Numerical experiments support the superior performance of …

Robust Inference for Convex Pairwise Difference Estimators
Matias D. Cattaneo, Michael Jansson, Kenichi Nagasawa
https://arxiv.org/abs/2510.05991 https://a…

Robust Inference for Convex Pairwise Difference Estimators
This paper develops distribution theory and bootstrap-based inference methods for a broad class of convex pairwise difference estimators. These estimators minimize a kernel-weighted convex-in-parameter function over observation pairs that are similar in terms of certain covariates, where the similarity is governed by a localization (bandwidth) parameter. While classical results establish asymptotic normality under restrictive bandwidth conditions, we show that valid Gaussian and bootstrap-based…

The weighted isoperimetric inequality and Sobolev inequality outside convex sets
Lu Chen, Jiali Lan
https://arxiv.org/abs/2510.01647 https://arxiv.org/pdf/…

The weighted isoperimetric inequality and Sobolev inequality outside convex sets
In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the $λ_w$-ABP method. The weight function $w$ is assumed to be positive, even, and homogeneous of degree $α$, such that $w^{1/α}$ is concave on $\R^n$. Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted Pólya-Szegö principle and a sharp weighted capillary Sobolev inequality outside convex …

Jensen convex functions and doubly stochastic matrices
Matyas Barczy, Zsolt P\'ales
https://arxiv.org/abs/2510.03715 https://arxiv.org/pdf/2510.03715…

Jensen convex functions and doubly stochastic matrices
Given an nxn doubly stochastic matrix P satisfying an appropriate condition of linear algebraic-type, and a function f defined on a nonempty interval, we show that the validity of a convexity-type functional inequality for f in terms P implies that f is Jensen convex. We also prove that if f is convex, then the functional inequality in question holds for all doubly stochastic matrices of any order. The particular case when the doubly stochastic matrix is a circulant one is also considered.

Closed-form Single UAV-aided Emitter Localization and Trajectory Design Using Doppler and TOA Measurements
Samaneh Motie, Hadi Zayyani, Mohammad Salman, Hasan Abu Hilal
https://arxiv.org/abs/2510.01778

Closed-form Single UAV-aided Emitter Localization and Trajectory Design Using Doppler and TOA Measurements
In this paper, a single Unmanned-Aerial-Vehicle (UAV)-aided localization algorithm which uses both Doppler and Time of Arrival (ToA) measurements is presented. In contrast to Doppler-based localization algorithms which are based on non-convex functions, exploiting ToA measurements in a Least-Square (LS) Doppler-based cost function, leads to a quadratic convex function whose minimizer lies on a line. Utilizing the ToA measurements in addition to the linear equation of minimizer, a closed form so…

Upper semicontinuous valuations on convex functions of one variable
Fernanda M. Ba\^eta
https://arxiv.org/abs/2510.05796 https://arxiv.org/pdf/2510.05796…

Upper semicontinuous valuations on convex functions of one variable
A classification of upper semicontinuous, translation and dually epi-translation invariant valuations is established on the space of convex Lipschitz function on $\mathbb{R}$ with compact domain.

Neural Optimal Transport Meets Multivariate Conformal Prediction
Vladimir Kondratyev, Alexander Fishkov, Nikita Kotelevskii, Mahmoud Hegazy, Remi Flamary, Maxim Panov, Eric Moulines
https://arxiv.org/abs/2509.25444

Neural Optimal Transport Meets Multivariate Conformal Prediction
We propose a framework for conditional vector quantile regression (CVQR) that combines neural optimal transport with amortized optimization, and apply it to multivariate conformal prediction. Classical quantile regression does not extend naturally to multivariate responses, while existing approaches often ignore the geometry of joint distributions. Our method parametrizes the conditional vector quantile function as the gradient of a convex potential implemented by an input-convex neural network…

The pluricomplex Poisson kernel for convex finite type domains
Leandro Arosio, Filippo Bracci, Matteo Fiacchi
https://arxiv.org/abs/2509.26230 https://arxi…

The pluricomplex Poisson kernel for convex finite type domains
Given a bounded convex domain $D\subset \mathbb C^n$ of finite D'Angelo type and a boundary point $ξ\in \partial D$, we prove that the homogeneous complex Monge-Ampère equation $(dd^cu)^n=0$ possesses a continuous strictly negative solution $Ω_ξ$ that vanishes on $\partial D\setminus \{ξ\}$ and has a simple pole at $ξ$. We establish that $Ω_ξ(z)$ equals (up to sign) the normal derivative at $ξ$ of the pluricomplex Green function $G_z$, and its sublevel sets are the horospheres centered…

Smooth Quasar-Convex Optimization with Constraints
David Mart\'inez-Rubio
https://arxiv.org/abs/2510.01943 https://arxiv.org/pdf/2510.01943

Smooth Quasar-Convex Optimization with Constraints
Quasar-convex functions form a broad nonconvex class with applications to linear dynamical systems, generalized linear models, and Riemannian optimization, among others. Current nearly optimal algorithms work only in affine spaces due to the loss of one degree of freedom when working with general convex constraints. Obtaining an accelerated algorithm that makes nearly optimal $\widetilde{O}(1/(γ\sqrtε))$ first-order queries to a $γ$-quasar convex smooth function \emph{with constraints} was i…

Relaxation of quasi-convex functionals with variable exponent growth
Giacomo Bertazzoni, Petteri Harjulehto, Peter H\"ast\"o, Elvira Zappale
https://arxiv.org/abs/2510.04672

Relaxation of quasi-convex functionals with variable exponent growth
We prove a relaxation result for a quasi-convex bulk integral functional with variable exponent growth in a suitable space of bounded variation type. A key tool is a decomposition under mild assumptions of the energy into absolutely continuous and singular parts weighted via a recession function.

Variational formulae for entropy-like functionals for states in von Neumann algebras
Andrzej {\L}uczak, Hanna Pods\k{e}dkowska, Rafa{\l} Wieczorek
https://arxiv.org/abs/2510.07605

Variational formulae for entropy-like functionals for states in von Neumann algebras
The paper presents variational formulae for entropy-like functionals, including Segal and Rényi entropies, for normal states on semifinite von Neumann algebras. The considered functionals are of the form $τ(f(h))$ where $τ$ is a normal faithful semifinite trace on this algebra, $h$ is a positive selfadjoint operator from $L^1(\M,τ)$, and $f$ is an appropriate convex or concave function. The results cover both finite and semifinite algebras, and the obtained formulae generalise known results…

Integrable Billiards and Related Topics
Misha Bialy, Andrey E. Mironov
https://arxiv.org/abs/2510.03790 https://arxiv.org/pdf/2510.03790

Integrable Billiards and Related Topics
This paper surveys our results on integrable billiards. We consider various models of billiards, including Birkhoff, outer, magnetic, and Minkowski billiards. Also, we discuss wire billiards and billiards in cones. For four models of convex plane billiards, we also discuss an isoperimetric-type inequality for the Mather $β$-function. We conclude with a section of open questions on this subject.

Approximate Bregman proximal gradient algorithm with variable metric Armijo--Wolfe line search
Kiwamu Fujiki, Shota Takahashi, Akiko Takeda
https://arxiv.org/abs/2510.06615 http…

Approximate Bregman proximal gradient algorithm with variable metric Armijo--Wolfe line search
We propose a variant of the approximate Bregman proximal gradient (ABPG) algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function. Although ABPG is known to converge globally to a stationary point even when the smooth part of the objective function lacks globally Lipschitz continuous gradients, and its iterates can often be expressed in closed form, ABPG relies on an Armijo line search to guarantee global convergence. Such reliance can slow down performanc…

On Hardy spaces, univalent functions and the second coefficient
Martin Chuaqui, Iason Efraimidis, Rodrigo Hern\'andez
https://arxiv.org/abs/2510.05395 https://

On Hardy spaces, univalent functions and the second coefficient
We consider normalized univalent functions with prescribed second Taylor coefficient $a_2$. For convex functions $f$ we study the Hardy spaces to which $f$ and $f'$ belong, refining in particular on a theorem of Eenigenburg and Keogh, and give a sharp asymptotic estimate and an explicit uniform bound for their coefficients. Relating the lower order of a convex function to the angle at infinity of its range we deduce that its range lies always in some sector of aperture $|a_2|π$. We g…

Characterizing Maximal Monotone Operators with Unique Representation
Sotiris Armeniakos, Aris Daniilidis
https://arxiv.org/abs/2510.09368 https://arxiv.org…

Characterizing Maximal Monotone Operators with Unique Representation
We study maximal monotone operators $A : X \rightrightarrows X^*$ whose Fitzpatrick family reduces to a singleton; such operators will be called uniquely representable. We show that every such operator is cyclically monotone (hence, $A=\partial f$ for some convex function $f$) if and only if it is 3-monotone. In Radon-Nikodým spaces, under mild conditions (which become superfluous in finite dimensions), we prove that a subdifferential operator $A=\partial f$ is uniquely representable if and on…

Long-Time Analysis of Stochastic Heavy Ball Dynamics for Convex Optimization and Monotone Equations
Radu Ioan Bot, Chiara Schindler
https://arxiv.org/abs/2510.02951 https://

Long-Time Analysis of Stochastic Heavy Ball Dynamics for Convex Optimization and Monotone Equations
In a separable real Hilbert space, we study the problem of minimizing a convex function with Lipschitz continuous gradient in the presence of noisy evaluations. To this end, we associate a stochastic Heavy Ball system, incorporating a friction coefficient, with the optimization problem. We establish existence and uniqueness of trajectory solutions for this system. Under a square integrability condition for the diffusion term, we prove almost sure convergence of the trajectory process to an opti…

An inverse problem for the Monge-Amp\`ere equation
Tony Liimatainen, Yi-Hsuan Lin
https://arxiv.org/abs/2510.11572 https://arxiv.org/pdf/2510.11572

An inverse problem for the Monge-Ampère equation
We extend the study of inverse boundary value problems to the setting of fully nonlinear PDEs by considering an inverse source problem for the Monge-Ampère equation \[ \det D^2 u = F. \] We prove that, on a convex Euclidean domain in the plane, the associated Dirichlet-to-Neumann (DN) map uniquely determines a positive source function $F$. The proof relies on recovering the Hessian of a solution to the equation, which is interpreted as a Riemannian metric $g$. Interestingly, although t…

New M-estimator of the leading principal component
Joni Virta, Una Radojicic, Marko Voutilainen
https://arxiv.org/abs/2510.02799 https://arxiv.org/pdf/2510…

New M-estimator of the leading principal component
We study the minimization of the non-convex and non-differentiable objective function $v \mapsto \mathrm{E} ( \| X - v \| \| X + v \| - \| X \|^2 )$ in $\mathbb{R}^p$. In particular, we show that its minimizers recover the first principal component direction of elliptically symmetric $X$ under specific conditions. The stringency of these conditions is studied in various scenarios, including a diverging number of variables $p$. We establish the consistency and asymp…

Convergence analysis of inexact MBA method for constrained upper-$\mathcal{C}^2$ optimization problems
Ruyu Liu, Shaohua Pan
https://arxiv.org/abs/2511.09940 https://arxiv.org/pdf/2511.09940 https://arxiv.org/html/2511.09940
arXiv:2511.09940v1 Announce Type: new
Abstract: This paper concerns a class of constrained optimization problems in which, the objective and constraint functions are both upper-$\mathcal{C}^2$. For such nonconvex and nonsmooth optimization problems, we develop an inexact moving balls approximation (MBA) method by a workable inexactness criterion for the solving of subproblems. By leveraging a global error bound for the strongly convex program associated with parametric optimization problems, we establish the full convergence of the iterate sequence under the partial bounded multiplier property (BMP) and the Kurdyka-{\L}ojasiewicz (KL) property of the constructed potential function, and achieve the local convergence rate of the iterate and objective value sequences if the potential function satisfies the KL property of exponent $q\in[1/2,1)$. A verifiable condition is also provided to check whether the potential function satisfies the KL property of exponent $q\in[1/2,1)$ at the given critical point. To the best of our knowledge, this is the first implementable inexact MBA method with a full convergence certificate for the constrained nonconvex and nonsmooth optimization problem.
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A Nonstationary Ruelle-Perron-Frobenius Theorem
Vaughn Climenhaga, Gregory Hemenway
https://arxiv.org/abs/2509.25467 https://arxiv.org/pdf/2509.25467

A Nonstationary Ruelle-Perron-Frobenius Theorem
The Ruelle-Perron-Frobenius (RPF) theorem is a powerful tool in the study of equilibrium measures and their statistical properties. We prove a nonstationary version of this theorem under general conditions involving an invariant sequence of real convex cones in function space.

Exponential convergence of a distributed divide-and-conquer algorithm for constrained convex optimization on networks
Nazar Emirov, Guohui Song, Qiyu Sun
https://arxiv.org/abs/2510.01511

Exponential convergence of a distributed divide-and-conquer algorithm for constrained convex optimization on networks
We propose a divide-and-conquer (DAC) algorithm for constrained convex optimization over networks, where the global objective is the sum of local objectives attached to individual agents. The algorithm is fully distributed: each iteration solves local subproblems around selected fusion centers and coordinates only with neighboring fusion centers. Under standard assumptions of smoothness, strong convexity, and locality on the objective function, together with polynomial growth conditions on the …

On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$
Thomas Pawlaschyk
https://arxiv.org/abs/2510.05009 https://arxi…

On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$
In the spirit of Lelong and Bochner, we show that an upper semi-continuous function defined on a open tube set $Ω=ω+ i\mathbb{R}^n$ in $\mathbb{C}^n$, where $ω$ is an open set in $\mathbb{R}^n$, and which is invariant in its imaginary part, is $q$-plurisubharmonic on $Ω$ (in the sense of Hunt and Murray) if and only if it is real $q$-convex on $ω$, i.e., it admits the local maximum property with respect to affine linear functions on real $(q+1)$-dimensional affine subspaces. From this, we …

H\"{o}lder property of the resolvent of a monotone operator in Banach spaces
Changchi Huang, Jigen Peng, Yuchao Tang
https://arxiv.org/abs/2510.03774 https://

Hölder property of the resolvent of a monotone operator in Banach spaces
Let $E$ be a Banach space, and let $J: E \to E^{*}$ denote the normalized duality mapping. In this paper, we establish an upper bound for $\|Jx - Jy\|$ in $q$-uniformly smooth Banach spaces, where the bound is expressed in terms of a relatively simple function of $\|x - y\|$. Subsequently, we derive the Hölder property of mappings of firmly nonexpansive type in 2-uniformly convex and $q$-uniformly smooth Banach spaces ($1

Locally Linear Convergence for Nonsmooth Convex Optimization via Coupled Smoothing and Momentum
Reza Rahimi Baghbadorani, Sergio Grammatico, Peyman Mohajerin Esfahani
https://arxiv.org/abs/2511.10239 https://arxiv.org/pdf/2511.10239 https://arxiv.org/html/2511.10239
arXiv:2511.10239v1 Announce Type: new
Abstract: We propose an adaptive accelerated smoothing technique for a nonsmooth convex optimization problem where the smoothing update rule is coupled with the momentum parameter. We also extend the setting to the case where the objective function is the sum of two nonsmooth functions. With regard to convergence rate, we provide the global (optimal) sublinear convergence guarantees of O(1/k), which is known to be provably optimal for the studied class of functions, along with a local linear rate if the nonsmooth term fulfills a so-call locally strong convexity condition. We validate the performance of our algorithm on several problem classes, including regression with the l1-norm (the Lasso problem), sparse semidefinite programming (the MaxCut problem), Nuclear norm minimization with application in model free fault diagnosis, and l_1-regularized model predictive control to showcase the benefits of the coupling. An interesting observation is that although our global convergence result guarantees O(1/k) convergence, we consistently observe a practical transient convergence rate of O(1/k^2), followed by asymptotic linear convergence as anticipated by the theoretical result. This two-phase behavior can also be explained in view of the proposed smoothing rule.
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S-D-RSM: Stochastic Distributed Regularized Splitting Method for Large-Scale Convex Optimization Problems
Maoran Wang, Xingju Cai, Yongxin Chen
https://arxiv.org/abs/2511.10133 https://arxiv.org/pdf/2511.10133 https://arxiv.org/html/2511.10133
arXiv:2511.10133v1 Announce Type: new
Abstract: This paper investigates the problems large-scale distributed composite convex optimization, with motivations from a broad range of applications, including multi-agent systems, federated learning, smart grids, wireless sensor networks, compressed sensing, and so on. Stochastic gradient descent (SGD) and its variants are commonly employed to solve such problems. However, existing algorithms often rely on vanishing step sizes, strong convexity assumptions, or entail substantial computational overhead to ensure convergence or obtain favorable complexity. To bridge the gap between theory and practice, we integrate consensus optimization and operator splitting techniques (see Problem Reformulation) to develop a novel stochastic splitting algorithm, termed the \emph{stochastic distributed regularized splitting method} (S-D-RSM). In practice, S-D-RSM performs parallel updates of proximal mappings and gradient information for only a randomly selected subset of agents at each iteration. By introducing regularization terms, it effectively mitigates consensus discrepancies among distributed nodes. In contrast to conventional stochastic methods, our theoretical analysis establishes that S-D-RSM achieves global convergence without requiring diminishing step sizes or strong convexity assumptions. Furthermore, it achieves an iteration complexity of $\mathcal{O}(1/\epsilon)$ with respect to both the objective function value and the consensus error. Numerical experiments show that S-D-RSM achieves up to 2--3$\times$ speedup compared to state-of-the-art baselines, while maintaining comparable or better accuracy. These results not only validate the algorithm's theoretical guarantees but also demonstrate its effectiveness in practical tasks such as compressed sensing and empirical risk minimization.
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ProxSTORM -- A Stochastic Trust-Region Algorithm for Nonsmooth Optimization
Robert J. Baraldi, Aurya Javeed, Drew P. Kouri, Katya Scheinberg
https://arxiv.org/abs/2510.03187 htt…

ProxSTORM -- A Stochastic Trust-Region Algorithm for Nonsmooth Optimization
We develop a stochastic trust-region algorithm for minimizing the sum of a possibly nonconvex Lipschitz-smooth function that can only be evaluated stochastically and a nonsmooth, deterministic, convex function. This algorithm, which we call ProxSTORM, generalizes STORM [1, 2] -- a stochastic trust-region algorithm for the unconstrained optimization of smooth functions -- and the inexact deterministic proximal trust-region algorithm in [3]. We generalize and, in some cases, simplify problem assu…

The Non-Attainment Phenomenon in Robust SOCPs
Vinh Nguyen
https://arxiv.org/abs/2510.00318 https://arxiv.org/pdf/2510.00318

The Non-Attainment Phenomenon in Robust SOCPs
A fundamental theorem of linear programming states that a feasible linear program is solvable if and only if its objective function is copositive with respect to the recession cone of its feasible set. This paper demonstrates that this crucial guarantee does not extend to Second-Order Cone Programs (SOCPs), a workhorse model in robust and convex optimization. We construct and analyze a rigorous counterexample derived from a robust linear optimization problem with ellipsoidal uncertainty. The re…

Non-Euclidean Broximal Point Method: A Blueprint for Geometry-Aware Optimization
Kaja Gruntkowska, Peter Richt\'arik
https://arxiv.org/abs/2510.00823 https://

Non-Euclidean Broximal Point Method: A Blueprint for Geometry-Aware Optimization
The recently proposed Broximal Point Method (BPM) [Gruntkowska et al., 2025] offers an idealized optimization framework based on iteratively minimizing the objective function over norm balls centered at the current iterate. It enjoys striking global convergence guarantees, converging linearly and in a finite number of steps for proper, closed and convex functions. However, its theoretical analysis has so far been confined to the Euclidean geometry. At the same time, emerging trends in deep lear…

A first-order method for constrained nonconvex--nonconcave minimax problems under a local Kurdyka-{\L}ojasiewicz condition
Zhaosong Lu, Xiangyuan Wang
https://arxiv.org/abs/2510.01168

A first-order method for constrained nonconvex--nonconcave minimax problems under a local Kurdyka-Łojasiewicz condition
We study a class of constrained nonconvex--nonconcave minimax problems in which the inner maximization involves potentially complex constraints. Under the assumption that the inner problem of a novel lifted minimax problem satisfies a local Kurdyka-Łojasiewicz (KL) condition, we show that the maximal function of the original problem enjoys a local Hölder smoothness property. We also propose a sequential convex programming (SCP) method for solving constrained optimization problems and establis…