arXiv subjects

M. V. Dolgopolik

Publications and source records attributed to M. V. Dolgopolik.

36 records · Page 2Linked to original sources

arXiv2021

Constrained Nonsmooth Problems of the Calculus of Variations

The paper is devoted to an analysis of optimality conditions for nonsmooth multidimensional problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary and isoperimetric constraints. To derive optimality conditions, we study generalised concepts of differentiability of nonsmooth functions called codifferentiability and quasidifferentiability. Under some natural and easily verifiable assumptions we prove that a nonsmooth integral functional defined on the Sobolev space is continuously codifferentiable and compute its codifferential and quasidifferential. Then we apply general optimality conditions for nonsmooth optimisation problems in Banach spaces to obtain optimality conditions for nonsmooth problems of the calculus of variations. Through a series of simple examples we demonstrate that our optimality conditions are sometimes better than existing ones in terms of various subdifferentials, in the sense that our optimality conditions can detect the non-optimality of a given point, when subdifferential-based optimality conditions fail to disqualify this point as non-optimal.

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arXiv2021

The Alternating Direction Method of Multipliers for Finding the Distance between Ellipsoids

We study several versions of the alternating direction method of multipliers (ADMM) for solving the convex problem of finding the distance between two ellipsoids and the nonconvex problem of finding the distance between the boundaries of two ellipsoids. In the convex case we present the ADMM with and without automatic penalty updates and demonstrate via numerical experiments on problems of various dimensions that our methods significantly outperform all other existing methods for finding the distance between ellipsoids. In the nonconvex case we propose a heuristic rule for updating the penalty parameter and a heuristic restarting procedure (a heuristic choice of a new starting for point for the second run of the algorithm). The restarting procedure was verified numerically with the use of a global method based on KKT optimality conditions. The results of numerical experiments on various test problems showed that this procedure always allows one to find a globally optimal solution in the nonconvex case. Furthermore, the numerical experiments also demonstrated that our version of the ADMM significantly outperforms existing methods for finding the distance between the boundaries of ellipsoids on problems of moderate and high dimensions.

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arXiv2020

Metric Regularity of Quasidifferentiable Mappings and Optimality Conditions for Nonsmooth Mathematical Programming Problems

This article is devoted to the analysis of necessary and/or sufficient conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova quasidifferentials. We obtain new necessary and sufficient conditions for the local metric regularity of a multifunction in terms of quasidifferentials of the distance function to this multifunction. We also propose a new MFCQ-type constraint qualification for a parametric system of quasidifferentiable equality and inequality constraints and prove that it ensures the metric regularity of a multifunction associated with this system. As an application, we utilize our constraint qualification to strengthen existing optimality conditions for quasidifferentiable programming problems with equality and inequality constraints. We also prove the independence of the optimality conditions of the choice of quasidifferentials and present a simple example in which the optimality conditions in terms of quasidifferentials detect the non-optimality of a given point, while optimality conditions in terms of various subdifferentials fail to disqualify this point as non-optimal.

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arXiv2020

A New Constraint Qualification and Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials

The paper is devoted to an analysis of a new constraint qualification and a derivation of the strongest existing optimality conditions for nonsmooth mathematical programming problems with equality and inequality constraints in terms of Demyanov-Rubinov-Polyakova quasidifferentials under the minimal possible assumptions. To this end, we obtain a novel description of convex subcones of the contingent cone to a set defined by quasidifferentiable equality and inequality constraints with the use of a new constraint qualification. We utilize these description and constraint qualification to derive the strongest existing optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials under less restrictive assumptions than in previous studies. The main feature of the new constraint qualification and related optimality conditions is the fact that they depend on individual elements of quasidifferentials of the objective function and constraints and are not invariant with respect to the choise of quasidifferentials. To illustrate the theoretical results, we present two simple examples in which optimality conditions in terms of various subdifferentials (in fact, any outer semicontinuous/limiting subdifferential) are satisfied at a nonoptimal point, while the optimality conditions obtained in this paper do not hold true at this point, that is, optimality conditions in terms of quasidifferentials, unlike the ones in terms of subdifferentials, detect the nonoptimality of this point.

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arXiv2020

A Unified Study of Necessary and Sufficient Optimality Conditions for Minimax and Chebyshev Problems with Cone Constraints

We present a unified study of first and second order necessary and sufficient optimality conditions for minimax and Chebyshev optimisation problems with cone constraints. First order optimality conditions for such problems can be formulated in several different forms: in terms of a linearised problem, in terms of Lagrange multipliers (KKT-points), in terms of subdifferentials and normal cones, in terms of a nonsmooth penalty function, in terms of cadres with positive cadre multipliers, and in an alternance form. We describe interconnections between all these forms of necessary and sufficient optimality conditions and prove that seemingly different conditions are in fact equivalent. We also demonstrate how first order optimality conditions can be reformulated in a more convenient form for particular classes of cone constrained optimisation problems and extend classical second order optimality condition for smooth cone constrained problems to the case of minimax and Chebyshev problems with cone constraints. The optimality conditions obtained in this article open a way for a development of new efficient structure-exploiting methods for solving cone constrained minimax and Chebyshev problems.

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arXiv2020

Exact Penalty Functions for Optimal Control Problems II: Exact Penalisation of Terminal and Pointwise State Constraints

The second part of our study is devoted to an analysis of the exactness of penalty functions for optimal control problems with terminal and pointwise state constraints. We demonstrate that with the use of the exact penalty function method one can reduce fixed-endpoint problems for linear time-varying systems and linear evolution equations with convex constraints on the control inputs to completely equivalent free-endpoint optimal control problems, if the terminal state belongs to the relative interior of the reachable set. In the nonlinear case, we prove that a local reduction of fixed-endpoint and variable-endpoint problems to equivalent free-endpoint ones is possible under the assumption that the linearised system is completely controllable, and point out some general properties of nonlinear systems under which a global reduction to equivalent free-endpoint problems can be achieved. In the case of problems with pointwise state inequality constraints, we prove that such problems for linear time-varying systems and linear evolution equations with convex state constraints can be reduced to equivalent problems without state constraints, provided one uses the $L^{\infty}$ penalty term, and Slater's condition holds true, while for nonlinear systems a local reduction is possible, if a natural constraint qualification is satisfied. Finally, we show that the exact $L^p$-penalisation of state constraints with finite $p$ is possible for convex problems, if Lagrange multipliers corresponding to the state constraints belong to $L^{p'}$, where $p'$ is the conjugate exponent of $p$, and for general nonlinear problems, if the cost functional does not depend on the control inputs explicitly.

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arXiv2019

New global optimality conditions for nonsmooth DC optimization problems

In this article we propose a new approach to an analysis of DC optimization problems. This approach was largely inspired by codifferential calculus and the method of codifferential descent and is based on the use of a so-called affine support set of a convex function instead of the Frenchel conjugate function. With the use of affine support sets we define a global codifferential mapping of a DC function and derive new necessary and sufficient global optimality conditions for DC optimization problems. We also provide new simple necessary and sufficient conditions for the global exactness of the $\ell_1$ penalty function for DC optimization problems with equality and inequality constraints and present a series of simple examples demonstrating a constructive nature of the new global optimality conditions. These examples show that when the optimality conditions are not satisfied, they can be easily utilised in order to find "global descent" directions of both constrained and unconstrained problems. As an interesting theoretical example, we apply our approach to the analysis of a nonsmooth problem of Bolza.

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arXiv2019

Exact Penalty Functions for Optimal Control Problems I: Main Theorem and Free-Endpoint Problems

In this two-part study we develop a general approach to the design and analysis of exact penalty functions for various optimal control problems, including problems with terminal and state constraints, problems involving differential inclusions, as well as optimal control problems for linear evolution equations. This approach allows one to simplify an optimal control problem by removing some (or all) constraints of this problem with the use of an exact penalty function, thus allowing one to reduce optimal control problems to equivalent variational problems, apply numerical methods for solving, e.g. problems without state constraints, to problems including such constraints, etc. In the first part of our study we strengthen some existing results on exact penalty functions for optimisation problems in infinite dimensional spaces and utilise them to study exact penalty functions for free-endpoint optimal control problems, which reduce these problems to equivalent variational ones. We also prove several auxiliary results on integral functionals and Nemytskii operators that are helpful for verifying the assumptions under which the proposed penalty functions are exact.

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arXiv2019

A convergence analysis of the method of codifferential descent

This paper is devoted to a detailed convergence analysis of the method of codifferential descent (MCD) developed by professor V.F. Demyanov for solving a large class of nonsmooth nonconvex optimization problems. We propose a generalization of the MCD that is more suitable for applications than the original method, and that utilizes only a part of a codifferential on every iteration, which allows one to reduce the overall complexity of the method. With the use of some general results on uniformly codifferentiable functions obtained in this paper, we prove the global convergence of the generalized MCD in the infinite dimensional case. Also, we propose and analyse a quadratic regularization of the MCD, which is the first general method for minimizing a codifferentiable function over a convex set. Apart from convergence analysis, we also discuss the robustness of the MCD with respect to computational errors, possible step size rules, and a choice of parameters of the algorithm. In the end of the paper we estimate the rate of convergence of the MCD for a class of nonsmooth nonconvex functions that arises, in particular, in cluster analysis. We prove that under some general assumptions the method converges with linear rate, and it convergence quadratically, provided a certain first order sufficient optimality condition holds true.

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arXiv2019

New Direct Numerical Methods for Some Multidimensional Problems of the Calculus of Variations

In this paper we develop a new approach to the design of direct numerical methods for multidimensional problems of the calculus of variations. The approach is based on a transformation of the problem with the use of a new class of Sobolev-like spaces that is studied in the article. This transformation allows one to analytically compute the direction of steepest descent of the main functional of the calculus of variations with respect to a certain inner product, and, in turn, to construct new direct numerical methods for multidimensional problems of the calculus of variations. In the end of the paper we point out how the approach developed in the article can be extended to the case of problems with more general boundary conditions, problems for functionals depending on higher order derivatives, and problems with isoperimetric and/or pointwise constraints.

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arXiv2018

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness

In the second part of our study we introduce the concept of global extended exactness of penalty and augmented Lagrangian functions, and derive the localization principle in the extended form. The main idea behind the extended exactness consists in an extension of the original constrained optimization problem by adding some extra variables, and then construction of a penalty/augmented Lagrangian function for the extended problem. This approach allows one to design extended penalty/augmented Lagrangian functions having some useful properties (such as smoothness), which their counterparts for the original problem might not possess. In turn, the global exactness of such extended merit functions can be easily proved with the use of the localization principle presented in this paper, which reduces the study of global exactness to a local analysis of a merit function based on sufficient optimality conditions and constraint qualifications. We utilize the localization principle in order to obtain simple necessary and sufficient conditions for the global exactness of the extended penalty function introduced by Huyer and Neumaier, and in order to construct a globally exact continuously differentiable augmented Lagrangian function for nonlinear semidefinite programming problems.

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arXiv2018

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

In this two-part study we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for finite-dimensional constrained optimization problems. This approach allows one to verify in a simple and straightforward manner whether a given penalty/augmented Lagrangian function is exact, i.e. whether the problem of unconstrained minimization of this function is equivalent (in some sense) to the original constrained problem, provided the penalty parameter is sufficiently large. Our approach is based on the so-called localization principle that reduces the study of global exactness to a local analysis of a chosen merit function near globally optimal solutions. In turn, such local analysis can usually be performed with the use of sufficient optimality conditions and constraint qualifications. In the first paper we introduce the concept of global parametric exactness and derive the localization principle in the parametric form. With the use of this version of the localization principle we recover existing simple necessary and sufficient conditions for the global exactness of linear penalty functions, and for the existence of augmented Lagrange multipliers of Rockafellar-Wets' augmented Lagrangian. Also, we obtain completely new necessary and sufficient conditions for the global exactness of general nonlinear penalty functions, and for the global exactness of a continuously differentiable penalty function for nonlinear second-order cone programming problems. We briefly discuss how one can construct a continuously differentiable exact penalty function for nonlinear semidefinite programming problems, as well.

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arXiv2018

Speed-Gradient Control of the Brockett Integrator

A nonsmooth extension of the speed-gradient algorithms in finite form is proposed. The conditions ensuring control goal (convergence of the goal function to zero) are established. A new algorithm is applied to almost global stabilization of the Brockett integrator that has become a popular benchmark for nonsmooth and discontinuous algorithms. It is proved that the designed control law stabilizes the Brockett integrator for any initial point that does not lie on the x3-axis. Besides, it is shown that the speed-gradient algorithm ensures stabilization with an arbitrarily small control level. An important feature of the proposed control is the fact that it is continuous along trajectories of the closed-loop system.

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arXiv2018

Abstract convex approximations of nonsmooth functions

In this article we utilise abstract convexity theory in order to unify and generalize many different concepts from nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality conditions for nonsmooth nonconvex optimization problems with the abstract codifferentiable or abstract quasidifferentiable objective function and constraints. Then we demonstrate how these conditions can be transformed into simpler and more constructive conditions in some particular cases.

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arXiv2018

A unifying theory of exactness of linear penalty functions II: parametric penalty functions

In this article we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.

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arXiv2018

A Unifying Theory of Exactness of Linear Penalty Functions

In this article we develop a theory of exact linear penalty functions that generalizes and unifies most of the results on exact penalization existing in the literature. We discuss several approaches to the study of both locally and globally exact linear penalty functions, and obtain various necessary and sufficient conditions for the exactness of a linear penalty function. We pay more attention than usual to necessary conditions that allows us to deeper understand the exact penalty technique.

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arXiv2018

Smooth Exact Penalty Functions: A General Approach

In this article we present a new perspective on the smooth exact penalty function proposed by Huyer and Neumaier that is becoming more and more popular tool for solving constrained optimization problems. Our approach to Huyer and Neumaier's exact penalty function allows one to apply previously unused tools (namely, parametric optimization) to the study of the exactness of this function. We give a new simple proof of the local exactness of Huyer and Neumair's penalty function that significantly generalizes all similar results existing in the literature. We also obtain new necessary and sufficient conditions for the global exactness of this penalty function.

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arXiv2018

Smooth exact penalty functions II: a reduction to standard exact penalty functions

A new class of smooth exact penalty functions was recently introduced by Huyer and Neumaier. In this paper, we prove that the new smooth penalty function for a constrained optimization problem is exact if and only if the standard nonsmooth penalty function for this problem is exact. We also provide some estimates of the exact penalty parameter of the smooth penalty function, and, in particular, show that it asymptotically behaves as the square of the exact penalty parameter of the standard $\ell_1$ penalty function. We briefly discuss a simple way to reduce the exact penalty parameter of the smooth penalty function, and study the effect of nonlinear terms on the exactness of this function.

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