arXiv subjects

Boris S. Mordukhovich

Publications and source records attributed to Boris S. Mordukhovich.

At least 37 records · Page 2Linked to original sources

arXiv2025

Second-Order Subdifferential Optimality Conditions in Nonsmooth Optimization

The paper is devoted to deriving novel second-order necessary and sufficient optimality conditions for local minimizers in rather general classes of nonsmooth unconstrained and constrained optimization problems in finite-dimensional spaces. The established conditions are expressed in terms of second-order subdifferentials of lower semicontinuous functions and mainly concern prox-regular objectives that cover a large territory in nonsmooth optimization and its applications. Our tools are based on the machinery of variational analysis and second-order generalized differentiation. The obtained general results are applied to problems of nonlinear programming, where the derived second-order optimality conditions are new even for problems with twice continuously differential data, being expressed there in terms of the classical Hessian matrices.

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arXiv2025

Generalized Twice Differentiability and Quadratic Bundles in Second-Order Variational Analysis

In this paper, we investigate the concepts of generalized twice differentiability and quadratic bundles of nonsmooth functions that have been very recently proposed by Rockafellar in the framework of second-order variational analysis. These constructions, in contrast to second-order subdifferentials, are defined in primal spaces. We develop new techniques to study generalized twice differentiability for a broad class of prox-regular functions, establish their novel characterizations. Subsequently, quadratic bundles of prox-regular functions are shown to be nonempty, which provides the ground of potential applications in variational analysis and optimization.

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arXiv2024

An Inexact Boosted Difference of Convex Algorithm for Nondifferentiable Functions

In this paper, we introduce an inexact approach to the Boosted Difference of Convex Functions Algorithm (BDCA) for solving nonconvex and nondifferentiable problems involving the difference of two convex functions (DC functions). Specifically, when the first DC component is differentiable and the second may be nondifferentiable, BDCA utilizes the solution from the subproblem of the DC Algorithm (DCA) to define a descent direction for the objective function. A monotone linesearch is then performed to find a new point that improves the objective function relative to the subproblem solution. This approach enhances the performance of DCA. However, if the first DC component is nondifferentiable, the BDCA direction may become an ascent direction, rendering the monotone linesearch ineffective. To address this, we propose an Inexact nonmonotone Boosted Difference of Convex Algorithm (InmBDCA). This algorithm incorporates two main features of inexactness: First, the subproblem therein is solved approximately allowing us for a controlled relative error tolerance in defining the linesearch direction. Second, an inexact nonmonotone linesearch scheme is used to determine the step size for the next iteration. Under suitable assumptions, we demonstrate that InmBDCA is well-defined, with any accumulation point of the sequence generated by InmBDCA being a critical point of the problem. We also provide iteration-complexity bounds for the algorithm. Numerical experiments show that InmBDCA outperforms both the nonsmooth BDCA (nmBDCA) and the monotone version of DCA in practical scenarios.

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arXiv2024

Fundamental Convergence Analysis of Sharpness-Aware Minimization

The paper investigates the fundamental convergence properties of Sharpness-Aware Minimization (SAM), a recently proposed gradient-based optimization method [Foret et al., 2021] that significantly improves the generalization of deep neural networks. The convergence properties, including the stationarity of accumulation points, the convergence of the sequence of gradients to the origin, the sequence of function values to the optimal value, and the sequence of iterates to the optimal solution, are established for the method. The universality of the provided convergence analysis, based on inexact gradient descent frameworks Khanh et al. [2023b], allows its extensions to efficient normalized versions of SAM such as F-SAM [Li et al., 2024], VaSSO [Li and Giannakis, 2023], RSAM [Liu et al., 2022], and to the unnormalized versions of SAM such as USAM [Andriushchenko and Flammarion, 2022]. Numerical experiments are conducted on classification tasks using deep learning models to confirm the practical aspects of our analysis.

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arXiv2024

Subdifferential Calculus for Ordered Set-Valued Mappings between Infinite-Dimensional Spaces

The paper is devoted to developing subdifferential theory for set-valued mappings taking values in ordered infinite-dimensional spaces. This study is motivated by applications to problems of vector and set optimization with various constraints in infinite dimensions. The main results establish new sum and chain rules for major subdifferential constructions associated with ordered set-valued mappings under appropriate qualification and sequentially normal compactness conditions.

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arXiv2024

Discrete approximations and optimality conditions for integro-differential inclusions

This paper addresses a new class of generalized Bolza problems governed by nonconvex integro-differential inclusions with endpoint constraints on trajectories, where the integral terms are given in the general (with time-dependent integrands in the dynamics) Volterra form. We pursue here a threefold goal. First we construct well-posed approximations of continuous-time integro-differential systems by their discrete-time counterparts with showing that any feasible solution to the original system can be strongly approximated in the $W^{1,2}$-norm topology by piecewise-linear extensions of feasible discrete trajectories. This allows us to verify in turn the strong convergence of discrete optimal solutions to a prescribed local minimizer for the original problem. Facing intrinsic nonsmoothness of original integro-differential problem and its discrete approximations, we employ appropriate tools of generalized differentiation in variational analysis to derive necessary optimality conditions for discrete-time problems (which is our second goal) and finally accomplish our third goal to obtain necessary conditions for the original continuous-time problems by passing to the limit from discrete approximations. In this way we establish, in particular, a novel necessary optimality condition of the Volterra type, which is the crucial result for dynamic optimization of integro-differential inclusions.

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arXiv2024

Coderivative-Based Newton Methods with Wolfe Linesearch for Nonsmooth Optimization

This paper introduces and develops novel coderivative-based Newton methods with Wolfe linesearch conditions to solve various classes of problems in nonsmooth optimization. We first propose a generalized regularized Newton method with Wolfe linesearch (GRNM-W) for unconstrained $C^{1,1}$ minimization problems (which are second-order nonsmooth) and establish global as well as local superlinear convergence of their iterates. To deal with convex composite minimization problems (which are first-order nonsmooth and can be constrained), we combine the proposed GRNM-W with two algorithmic frameworks: the forward-backward envelope and the augmented Lagrangian method resulting in the two new algorithms called CNFB and CNAL, respectively. Finally, we present numerical results to solve Lasso and support vector machine problems appearing in, e.g., machine learning and statistics, which demonstrate the efficiency of the proposed algorithms.

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arXiv2024

Optimal control of perturbed sweeping processes with applications to general robotics models

This paper primarily focuses on the practical applications of optimal control theory for perturbed sweeping processes within the realm of robotics dynamics. By describing these models as controlled sweeping processes with pointwise control and state constraints and by employing necessary optimality conditions for such systems, we formulate optimal control problems suitable to these models and develop numerical algorithms for their solving. Subsequently, we use the Python Dynamic Optimization library GEKKO to simulate solutions to the posed robotics problems in the case of any fixed number of robots under different initial conditions.

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arXiv2024

Relationships between Global and Local Monotonicity of Operators

The paper is devoted to establishing relationships between global and local monotonicity, as well as their maximality versions, for single-valued and set-valued mappings between finite-dimensional and infinite-dimensional spaces. We first show that for single-valued operators with convex domains in locally convex topological spaces, their continuity ensures that their global monotonicity agrees with the local one around any point of the graph. This also holds for set-valued mappings defined on the real line under a certain connectedness condition. The situation is different for set-valued operators in multidimensional spaces as demonstrated by an example of locally monotone operator on the plane that is not globally monotone. Finally, we invoke coderivative criteria from variational analysis to characterize both global and local maximal monotonicity of set-valued operators in Hilbert spaces to verify the equivalence between these monotonicity properties under the closed-graph and global hypomonotonicity assumptions.

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arXiv2024

A New Inexact Gradient Descent Method with Applications to Nonsmooth Convex Optimization

The paper proposes and develops a novel inexact gradient method (IGD) for minimizing C1-smooth functions with Lipschitzian gradients, i.e., for problems of C1,1 optimization. We show that the sequence of gradients generated by IGD converges to zero. The convergence of iterates to stationary points is guaranteed under the Kurdyka- Lojasiewicz (KL) property of the objective function with convergence rates depending on the KL exponent. The newly developed IGD is applied to designing two novel gradient-based methods of nonsmooth convex optimization such as the inexact proximal point methods (GIPPM) and the inexact augmented Lagrangian method (GIALM) for convex programs with linear equality constraints. These two methods inherit global convergence properties from IGD and are confirmed by numerical experiments to have practical advantages over some well-known algorithms of nonsmooth convex optimization.

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arXiv2024

Stability Criteria and Calculus Rules via Conic Contingent Coderivatives in Banach Spaces

This paper addresses the study of novel constructions of variational analysis and generalized differentiation that are appropriate for characterizing robust stability properties of constrained set-valued mappings/multifunctions between Banach spaces important in optimization theory and its applications. Our tools of generalized differentiation revolves around the newly introduced concept of $\varepsilon$-regular normal cone to sets and associated coderivative notions for set-valued mappings. Based on these constructions, we establish several characterizations of the central stability notion known as the relative Lipschitz-like property of set-valued mappings in infinite dimensions. Applying a new version of the constrained extremal principle of variational analysis, we develop comprehensive sum and chain rules for our major constructions of conic contingent coderivatives for multifunctions between appropriate classes of Banach spaces.

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arXiv2023

Local Minimizers of Nonconvex Functions in Banach Spaces via Moreau Envelopes

This paper investigates the preservation of local minimizers and strong minimizers of extended-real-valued lower semicontinuous functions under taking their Moreau envelopes. We address a general setting of Banach spaces, while all the obtained results are new even for functions in finite dimensions. Our main motivation came from applications to numerical methods in nonsmooth optimization dealing with broad classes of nondifferentiable and nonconvex functions. The paper also formulates and discusses some open questions stemming from this study.

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arXiv2023

General Derivative-Free Optimization Methods under Global and Local Lipschitz Continuity of Gradients

This paper addresses the study of derivative-free smooth optimization problems, where the gradient information on the objective function is unavailable. Two novel general derivative-free methods are proposed and developed for minimizing such functions with either global or local Lipschitz continuous gradients. The newly developed methods use gradient approximations based on finite differences, where finite difference intervals are automatically adapted to the magnitude of the exact gradients without knowing them exactly. The suggested algorithms achieve fundamental convergence results, including stationarity of accumulation points in general settings as well as global convergence with constructive convergence rates when the Kurdyka-Łojasiewicz property is imposed. The local convergence of the proposed algorithms to nonisolated local minimizers, along with their local convergence rates, is also analyzed under this property. Numerical experiences involving various convex, nonconvex, noiseless, and noisy functions demonstrate that the new methods exhibit essential advantages over other state-of-the-art methods in derivative-free optimization.

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arXiv2023

Optimal control of sweeping processes in unmanned surface vehicle and nanoparticle modeling

This paper addresses novel applications to practical modeling of the newly developed theory of necessary optimality conditions in controlled sweeping/Moreau processes with free time and pointwise control and state constraints. Problems of this type appear, in particular, in dynamical models dealing with unmanned surface vehicles (USVs) and nanoparticles. We formulate optimal control problems for a general class of such dynamical systems and show that the developed necessary optimality conditions for constrained free-time controlled sweeping processes lead us to designing efficient procedures to solve practical models of this class. Moreover, the paper contains numerical calculations of optimal solutions to marine USVs and nanoparticle models in specific situations. Overall, this study contributes to the advancement of optimal control theory for constrained sweeping processes and its practical applications in the fields of marine USVs and nanoparticle modeling.

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arXiv2023

Discrete Approximations and Optimality Conditions for Controlled Free-Time Sweeping Processes

The paper is devoted to the study of a new class of optimal control problems governed by discontinuous constrained differential inclusions of the sweeping type with involving the duration of the dynamic process into optimization. We develop a novel version of the method of discrete approximations of its own qualitative and numerical values with establishing its well-posedness and strong convergence to optimal solutions of the controlled sweeping process. Using advanced tools of first-order and second-order variational analysis and generalized differentiation allows us to derive new necessary conditions for optimal solutions of the discrete-time problems and then, by passing to the limit in the discretization procedure, for designated local minimizers in the original problem of sweeping optimal control. The obtained results are illustrated by a numerical example.

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arXiv2023

Variational and Strong Variational Convexity in Infinite-Dimensional Variational Analysis

This paper is devoted to a systematic study and characterizations of the fundamental notions of variational and strong variational convexity for lower semicontinuous functions. While these notions have been quite recently introduced by Rockafellar, the importance of them has been already recognized and documented in finite-dimensional variational analysis and optimization. Here we address general infinite-dimensional settings and derive comprehensive characterizations of both variational and strong variational convexity notions by developing novel techniques, which are essentially different from finite-dimensional counterparts. As a consequence of the obtained characterizations, we establish new quantitative and qualitative relationships between strong variational convexity and tilt stability of local minimizers in appropriate frameworks of Banach spaces.

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arXiv2023

Variational Convexity of Functions in Banach Spaces

This paper addresses the study and characterizations of variational convexity of extended-real-valued functions on Banach spaces. This notion has been recently introduced by Rockafellar, and its importance has been already realized and applied to continuous optimization problems in finite-dimensional spaces. Variational convexity in infinite-dimensional spaces, which is studied here for the first time, is significantly more involved and requires the usage of powerful tools of geometric functional analysis together with variational analysis and generalized differentiation in Banach spaces.

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arXiv2023

Local Maximal Monotonicity in Variational Analysis and Optimization

The paper is devoted to a systematic study and characterizations of notions of local maximal monotonicity and their strong counterparts for set-valued operators that appear in variational analysis, optimization, and their applications. We obtain novel resolvent characterizations of these notions together with efficient conditions for their preservation under summation in broad infinite-dimensional settings. Further characterizations of these notions are derived by using generalized differentiation of variational analysis in the framework of Hilbert spaces.

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