arXiv subjects

Boris S. Mordukhovich

Publications and source records attributed to Boris S. Mordukhovich.

At least 19 recordsLinked to original sources

arXiv2026

Optimization of a Perturbed Implicit Sweeping Process by Constrained Discontinuous Controls

This paper studies optimal control problems governed by an implicit sweeping process with velocity-dependent constraint geometry, where the dynamics are described by a differential inclusion involving the normal cone to a moving convex polyhedral set. Such systems form a challenging class of nonsmooth differential inclusions due to the implicit coupling between the state, velocity, and constraint reaction. To address these difficulties, we use a method of discrete approximations together with tools of variational analysis and generalized differentiation. This approach allows us to derive necessary optimality conditions for local minimizer in discrete-time problems and subsequently pass to the limit to obtain optimality conditions for the original continuous-time system. The obtained results extend existing optimal control theory for sweeping processes to a class of implicit dynamical systems with state-and velocity-dependent constraints.

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arXiv2026

Optimal Control of Prox-Regular Sweeping Processes with Variable Time

The paper is devoted to a broad class of optimal control problems governed by constrained sweeping processes with controlled dynamics, controlled prox-regular moving sets, and variable time included in the optimization process. Employing an advanced version of the method of discrete approximations, we verify its well-posedness with appropriate strong convergence of feasible and optimal solutions. This approach leads us to considering a parametric family of discrete-time controlled sweeping processes, which are reduced in turn to nondynamic mathematical programs with special types of geometric and functional constraints. Invoking powerful tools of first-order and second-order variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete-time and then---by passing to the limit over the diminishing step of discretization---continuous-time controlled sweeping processes. A crucial role in our device is played by explicit coderivative evaluations of set-valued mapping associated with the sweeping dynamics. In this way, we establish novel necessary optimality conditions for local minimizers of continuous-time controlled sweeping processes that contain absolutely continuous and BV adjoint arcs, local and global maximization conditions, $L^2$ active set multipliers, vector measures generated by sweeping and control constraints, and free-time transversality condition expressed through the averaged Hamiltonian. For suitably selected multipliers, we also obtain nondegenerate forms of these conditions when the optimal trajectory starts on the boundary of the moving set. The efficiency of the obtained results is illustrated by some examples related to practical modeling.

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arXiv2026

Nonmonotone Coderivative-Based Newton Methods for Nonsmooth Optimization with Machine Learning Applications

Newton-type algorithms are among the most effective methods for solving optimization problems because of their rapid local convergence. However, extending these methods to nonsmooth optimization is challenging due to the difficulty of incorporating second-order information. To address this issue, we propose a coderivative-based Newton framework for solving both unconstrained and constrained nonsmooth optimization problems using tools from variational analysis and generalized differentiation. The proposed method employs generalized Hessians, defined as coderivatives of the subgradient mapping, and is applicable to both $C ^{1,1}$ functions and convex composite optimization problems with extended-real-valued components. To enhance robustness, the algorithm incorporates an adaptive regularization strategy that enables it to handle problems whose generalized Hessians are positive semidefinite. In addition, a Hybrid Adaptive Nonmonotone (HAN) line search scheme is developed as a globalization technique to improve global convergence and practical performance. Under standard assumptions, both the exact and inexact versions of the algorithm are globally convergent and achieve local superlinear convergence when the associated subgradient mapping satisfies the semismooth* property. The proposed framework is further extended to convex composite optimization through the forward-backward envelope, allowing it to handle problems with or without strong convexity in the smooth component of the objective function. Numerical experiments on Lasso, logistic Lasso, and support vector machine (SVM) problems demonstrate the efficiency and robustness of the proposed algorithm. Comparisons with several well-established first-order and second-order methods for nonsmooth optimization show that the proposed approach is computationally competitive while maintaining strong convergence properties.

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arXiv2026

Inexact Proximal Point and Tseng Algorithms with Nonsummable Errors to Solve Monotone Inclusions

In this paper, we establish, for the first time in the literature, the convergence of the practical versions of the Inexact Proximal Point Algorithm (IPPA) and the Inexact Tseng Algorithm (ITA) for computing approximate solutions to monotone inclusions in Hilbert spaces under the the presence of nonsummable errors. Our ap- proach relies on Tikhonov regularization, the contraction property of the associated monotone operators, and the recently developed R-continuity theory. The proposed techniques and results can be extended to analyze various important inexact algorithms in optimization-related problems with nonsummable errors.

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arXiv2026

Convergence and Stability Analysis of a Generalized Proximal Point Algorithm and Its Inexact Version

In this paper, we introduce the notion of adaptive strong monotonicity and establish a linear convergence of the recently proposed Generalized Proximal Point Algorithm (GPPA) for computing approximate solutions to nonmonotone inclusion problems in Hilbert spaces. We also investigate the Inexact Generalized Proximal Point Algorithm (IGPPA) in the presence of nondiminishing errors.

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arXiv2026

Extended SQP Methods in Nonsmooth Difference Programming Applied to Problems with Variational Inequality Constraints

This paper explores a new class of constrained difference programming problems, where the objective and constraints are formulated as differences of functions, without requiring their convexity. To investigate such problems, novel variants of the extended sequential quadratic method are introduced. These algorithms iteratively solve strongly convex quadratic subproblems constructed via linear approximations of the given data by using their gradients and subgradients. The convergence of the proposed methods is rigorously analyzed by employing, in particular, the Polyak-Łojasiewicz-Kurdyka property that ensures global convergence for various classes of functions in the problem formulation, e.g., semialgebraic ones. The original framework is further extended to address difference programming problems with variational inequality (VI) constraints. By reformulating VI constraints via regularized gap functions, such problems are naturally embedded into constrained difference programming that leads us to direct applications of the proposed algorithms. Numerical experiments for the class of continuous network design problems demonstrate the efficiency of the new methods.

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arXiv2026

Coderivative-Based Newton Methods in Structured Nonconvex and Nonsmooth Optimization

This paper proposes and develops new Newton-type methods to solve structured nonconvex and nonsmooth optimization problems with justifying their fast local and global convergence by means of advanced tools of variational analysis and generalized differentiation. The objective functions belong to a broad class of prox-regular functions with specification to constrained optimization of nonconvex structured sums. We also develop a novel line search method, which is an extension of the proximal gradient algorithm while allowing us to globalize the proposed coderivative-based Newton methods by incorporating the machinery of forward-backward envelopes. Applications and numerical experiments, which are provided for nonconvex least squares regression models, Student's t-regression with an $\ell^0$-penalty, and image restoration problems, demonstrate the efficiency of the proposed algorithms.

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arXiv2026

Complete Characterizations of Well-Posedness in Parametric Composite Optimization

This paper characterizes the well-posedness of Karush-Kuhn-Tucker system for perturbed composite optimization. Using the parabolic regularity, we introduce a novel second-order variational function, shown to be the pivotal object governing second-order behavior. This foundational result yields the strong second-order sufficient condition introduced here for the general class of composite optimization problems to naturally extend the classical second-order sufficient condition in nonlinear programming. Then we obtain several equivalent characterizations of the second-order qualification condition (SOQC) and highlight its equivalence to the constraint nondegeneracy condition under the $\mathcal{C}^{2}$-cone reducibility assumption. These insights lead us to multiple equivalent conditions for the major Lipschitz-like/Aubin property of KKT systems, including SOQC combined with the new second-order subdifferential condition and SOQC combined with tilt stability of local minimizers. Under the $\mathcal{C}^{2}$-cone reducibility, we settle the long-standing question by proving the equivalence between the Aubin property of KKT systems around local minimizers and the classical notion of strong regularity under some additional assumptions. Finally, we demonstrate that the Lipschitz-like property is equivalent to the nonsingularity of the generalized Jacobian associated with the KKT system under a certain verifiable assumption. These results provide a unified and rigorous framework for analyzing stability and sensitivity of solutions to composite optimization problems, as well as for the design and justification of numerical algorithms.

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arXiv2026

Convergence Rates for the Alternating Minimization Algorithm in Structured Nonsmooth and Nonconvex Optimization

This paper is devoted to developing the alternating minimization algorithm for problems of structured nonconvex optimization proposed by Attouch, Bolté, Redont, and Soubeyran in 2010. Our main result provides significant improvements of the convergence rate of the algorithm, especially under the low exponent Polyak-Łojasiewicz-Kurdyka condition when we establish either finite termination of this algorithm or its superlinear convergence rate instead of the previously known linear convergence. We also investigate the PLK exponent calculus and discuss applications to noncooperative games and behavioral science.

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arXiv2025

New Location Science Models with Applications to UAV-Based Disaster Relief

Natural and human-made disasters can cause severe devastation and claim thousands of lives worldwide. Therefore, developing efficient methods for disaster response and management is a critical task for relief teams. One of the most essential components of effective response is the rapid collection of information about affected areas, damages, and victims. More data translates into better coordination, faster rescue operations, and ultimately, more lives saved. However, in some disasters, such as earthquakes, the communication infrastructure is often partially or completely destroyed, making it extremely difficult for victims to send distress signals and for rescue teams to locate and assist them in time. Unmanned Aerial Vehicles (UAVs) have emerged as valuable tools in such scenarios. In particular, a fleet of UAVs can be dispatched from a mobile station to the affected area to facilitate data collection and establish temporary communication networks. Nevertheless, real-world deployment of UAVs faces several challenges, with adverse weather conditions--especially wind--being among the most significant. To address this, we develop a novel mathematical framework to determine the optimal location of a mobile UAV station while explicitly accounting for the heterogeneity of the UAVs and the effect of wind. In particular, we generalize the Sylvester problem to introduce the Sylvester-Fermat-Torricelli (SFT) problem, which captures complex factors such as wind influence, UAV heterogeneity, and back-and-forth motion within a unified framework. The proposed framework enhances the practicality of UAV-based disaster response planning by accounting for real-world factors such as wind and UAV heterogeneity. Experimental results demonstrate that it can reduce wasted operational time by up to 84%, making post-disaster missions significantly more efficient and effective.

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arXiv2025

Compact R-Continuity with Applications to Solving Inclusions and Convergence of Algorithms

This paper investigates the notion of compact R-continuity and its specifications for set-valued mappings between Banach spaces. We reveal several important properties of compact R-continuity in general settings and show that in finite dimensions, this notion is supported by the classical Lojasiewicz inequality for analytic functions. An application of compact R-continuity and the obtained results is given to convergence analysis for a broad class of descent algorithms in nonsmooth optimization. We also show that this notion is instrumental for the design and justification of a novel R-class of algorithms to solve inclusion problems.

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arXiv2025

Second-Order Characterizations of Tilt Stability in Composite Optimization

Tilt stability is a fundamental concept of variational analysis and optimization that plays a pivotal role in both theoretical issues and numerical computations. This paper investigates tilt stability of local minimizers for a general class of composite optimization problems in finite dimensions, where extended-real-valued objectives are compositions of parabolically regular and smooth functions. Under the weakest metric subregularity constraint qualification and other verifiable conditions, we establish unified neighborhood and pointbased characterizations of tilt stability via second-order generalized differentiation. The obtained results provide a rigorous theoretical foundation for further developments on variational stability and numerical algorithms of optimization and related topics.

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arXiv2025

Characterizations of Strong Variational Sufficiency in General Models of Composite Optimization

This paper investigates a recently introduced notion of strong variational sufficiency in optimization problems whose importance has been highly recognized in optimization theory, numerical methods, and applications. We address a general class of composite optimization problems and establish complete characterizations of strong variational sufficiency for their local minimizers in terms of a generalized version of the strong second-order sufficient condition (SSOSC) and the positive-definiteness of an appropriate generalized Hessian of the augmented Lagrangian calculated at the point in question. The generalized SSOSC is expressed via a novel second-order variational function, which reflects specific features of nonconvex composite models. The imposed assumptions describe the spectrum of composite optimization problems covered by our approach while being constructively implemented for nonpolyhedral problems that involve the nuclear norm function and the indicator function of the positive-semidefinite cone without any constraint qualifications.

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arXiv2025

Relative Well-Posedness of Truncated Constrained Systems Accompanied by Variational Calculus

The paper concerns foundations of sensitivity and stability analysis in optimization and related areas, being primarily addressed truncated constrained systems. We consider general models, which are described by multifunctions between Banach spaces and concentrate on characterizing their well-posedness properties that revolve around Lipschitz stability and metric regularity relative to sets. Invoking tools of variational analysis and generalized differentiation, we introduce new robust notions of relative contingent coderivatives. The novel machinery of variational analysis leads us to establishing complete characterizations of such properties and developing basic rules of variational calculus interrelated with the obtained characterizations of well-posedness. Most of the our results valid in general infinite-dimensional settings are also new in finite dimensions.

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arXiv2025

The Least Singular Value Function in Variational Analysis

Metric regularity is among the central concepts of nonlinear and variational analysis, constrained optimization, and their numerous applications. However, metric regularity can be elusive for some important ill-posed classes of problems including polynomial equations, parametric variational systems, smooth reformulations of complementarity systems with degenerate solutions, etc. The study of stability issues for such problems can often not rely on the machinery of first-order variational analysis, and so higher-order regularity concepts have been proposed in recent years. In this paper, we investigate some notions of mixed-order regularity by using advanced tools of first-order and second-order variational analysis and generalized differentiation of both primal and dual types. Efficient characterizations of such mixed-order regularity concepts are established by employing a fresh notion of the least singular value function. The obtained conditions are applied to deriving constructive criteria for mixed-order regularity in coupled constraint and variational systems.

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arXiv2025

Optimal Control of Nonconvex Sweeping Processes with Variable Time via Finite-Difference Approximations

The paper is devoted to the study of a new class of optimal control problems for nonsmooth dynamical systems governed by nonconvex discontinuous differential inclusions of the sweeping type with involving variable time into optimization. We develop a novel version of the method of discrete approximations of its own qualitative and numerical importance with establishing its well-posedness and strong convergence to optimal solutions of the controlled sweeping process. Using advanced tools of variational analysis and generalized differentiation leads us to deriving new necessary conditions for optimal solutions to discrete approximation problems, which serve as suboptimality conditions for the original continuous-time controlled sweeping process. The obtained results are applied to a class of motion models of practical interest, where the established necessary conditions are used to investigate the agents' interactions and to develop an algorithm for calculating optimal solutions.

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arXiv2025

Regular Subgradients of Marginal Functions with Applications to Calculus and Bilevel Programming

The paper addresses the study and applications of a broad class of extended-real-valued functions, known as optimal value or marginal functions, which are frequently appeared in variational analysis, parametric optimization, and a variety of applications. Functions of this type are intrinsically nonsmooth and require the usage of tools of generalized differentiation. The main results of this paper provide novel evaluations and exact calculations of regular/Fréchet subgradients and their singular counterparts for general classes of marginal functions via their given data. The obtained results are applied to establishing new calculus rules for such subgradients and necessary optimality conditions in bilevel programming

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arXiv2025

Characterizations of Variational Convexity and Tilt Stability via Quadratic Bundles

In this paper, we establish characterizations of variational $s$-convexity and tilt stability for prox-regular functions in the absence of subdifferential continuity via quadratic bundles, a kind of primal-dual generalized second-order derivatives recently introduced by Rockafellar. Deriving such characterizations in the effective pointbased form requires a certain revision of quadratic bundles investigated below. Our device is based on the notion of generalized twice differentiability and its novel characterization via classical twice differentiability of the associated Moreau envelopes combined with various limiting procedures for functions and sets.

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