scholarly journals Penalty and Augmented Lagrangian Methods for Constrained DC Programming

2022 ◽  
Author(s):  
Zhaosong Lu ◽  
Zhe Sun ◽  
Zirui Zhou
Keyword(s):  
Stationary Point ◽  
Penalty Method ◽  
Augmented Lagrangian ◽  
Initial Point ◽  
Accumulation Point ◽  
Stationary Points ◽  
Augmented Lagrangian Methods ◽  
Smooth Functions ◽  
Lagrangian Multipliers ◽  
Solution Sequence

In this paper, we consider a class of structured nonsmooth difference-of-convex (DC) constrained DC programs in which the first convex component of the objective and constraints is the sum of a smooth and a nonsmooth function, and their second convex component is the supremum of finitely many convex smooth functions. The existing methods for this problem usually have a weak convergence guarantee or require a feasible initial point. Inspired by the recent work by Pang et al. [Pang J-S, Razaviyayn M, Alvarado A (2017) Computing B-stationary points of nonsmooth DC programs. Math. Oper. Res. 42(1):95–118.], in this paper, we propose two infeasible methods with a strong convergence guarantee for the considered problem. The first one is a penalty method that consists of finding an approximate D-stationary point of a sequence of penalty subproblems. We show that any feasible accumulation point of the solution sequence generated by such a penalty method is a B-stationary point of the problem under a weakest possible assumption that it satisfies a pointwise Slater constraint qualification (PSCQ). The second one is an augmented Lagrangian (AL) method that consists of finding an approximate D-stationary point of a sequence of AL subproblems. Under the same PSCQ condition as for the penalty method, we show that any feasible accumulation point of the solution sequence generated by such an AL method is a B-stationary point of the problem, and moreover, it satisfies a Karush–Kuhn–Tucker type of optimality condition for the problem, together with any accumulation point of the sequence of a set of auxiliary Lagrangian multipliers. We also propose an efficient successive convex approximation method for computing an approximate D-stationary point of the penalty and AL subproblems. Finally, some numerical experiments are conducted to demonstrate the efficiency of our proposed methods.

A LOCALLY SMOOTHING METHOD FOR MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

2015 ◽  
Vol 56 (3) ◽  
pp. 299-315 ◽  
Author(s):  
YU CHEN ◽  
ZHONG WAN
Keyword(s):  
Stationary Point ◽  
Sufficient Conditions ◽  
Accumulation Point ◽  
Smoothing Method ◽  
Stationary Points ◽  
Local Perturbation ◽  
Complementarity Constraints ◽  
Approximate Problem ◽  
Mathematical Programs ◽  
Strongly Stationary

We propose a locally smoothing method for some mathematical programs with complementarity constraints, which only incurs a local perturbation on these constraints. For the approximate problem obtained from the smoothing method, we show that the Mangasarian–Fromovitz constraints qualification holds under certain conditions. We also analyse the convergence behaviour of the smoothing method, and present some sufficient conditions such that an accumulation point of a sequence of stationary points for the approximate problems is a C-stationary point, an M-stationary point or a strongly stationary point. Numerical experiments are employed to test the performance of the algorithm developed. The results obtained demonstrate that our algorithm is much more promising than the similar ones in the literature.

scholarly journals Topological Approach to Mathematical Programs with Switching Constraints

2021 ◽  
Author(s):  
Vladimir Shikhman
Keyword(s):  
Stationary Point ◽  
Mountain Pass Theorem ◽  
Strong Stability ◽  
Complete Characterization ◽  
Stationary Points ◽  
Mathematical Programs ◽  
Linear Independence Constraint Qualification ◽  
Point Set ◽  
Stationary Level

AbstractWe study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian’s Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy and strong stability of W-stationary points. A complete characterization of strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, no bi-active Lagrange multipliers of a strongly stable W-stationary point can vanish.

scholarly journals On a Robust and Efficient Numerical Scheme for the Simulation of Stationary 3-Component Systems with Non-Negative Species-Concentration with an Application to the Cu Deposition from a Cu-(β-alanine)-Electrolyte

Algorithms ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 113
Author(s):  
Stephan Daniel Schwoebel ◽  
Thomas Mehner ◽  
Thomas Lampke
Keyword(s):  
Augmented Lagrangian ◽  
Computation Time ◽  
Species Distributions ◽  
Chemical Processes ◽  
Diffusion Reaction ◽  
Augmented Lagrangian Methods ◽  
Major Question ◽  
Component Systems ◽  
Reaction Models ◽  
Reaction Equations

Three-component systems of diffusion–reaction equations play a central role in the modelling and simulation of chemical processes in engineering, electro-chemistry, physical chemistry, biology, population dynamics, etc. A major question in the simulation of three-component systems is how to guarantee non-negative species distributions in the model and how to calculate them effectively. Current numerical methods to enforce non-negative species distributions tend to be cost-intensive in terms of computation time and they are not robust for big rate constants of the considered reaction. In this article, a method, as a combination of homotopy methods, modern augmented Lagrangian methods, and adaptive FEMs is outlined to obtain a robust and efficient method to simulate diffusion–reaction models with non-negative concentrations. Although in this paper the convergence analysis is not described rigorously, multiple numerical examples as well as an application to elctro-deposition from an aqueous Cu2+-(β-alanine) electrolyte are presented.

scholarly journals An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

2021 ◽  
Author(s):  
Christian Kanzow ◽  
Andreas B. Raharja ◽  
Alexandra Schwartz
Keyword(s):  
Constrained Optimization ◽  
Numerical Experiments ◽  
Penalty Method ◽  
Augmented Lagrangian ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Constrained Optimization Problems ◽  
Strongly Stationary ◽  
Type Constraint

AbstractA reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.

scholarly journals On Augmented Lagrangian Methods with General Lower-Level Constraints

2008 ◽  
Vol 18 (4) ◽  
pp. 1286-1309 ◽  
Author(s):  
R. Andreani ◽  
E. G. Birgin ◽  
J. M. Martínez ◽  
M. L. Schuverdt
Keyword(s):  
Augmented Lagrangian ◽  
Augmented Lagrangian Methods ◽  
Lagrangian Methods

Nonsmooth Optimization Method for Image Restoration

1995 ◽  
Author(s):  
Kazufumi Ito ◽  
Karl Kunisch
Keyword(s):  
Image Restoration ◽  
Nonsmooth Optimization ◽  
Numerical Optimization ◽  
Augmented Lagrangian ◽  
Energy Criterion ◽  
Optimization Methods ◽  
Optimization Method ◽  
Augmented Lagrangian Methods ◽  
Active Set ◽  
First Order

Abstract In this paper we discuss applications of the numerical optimization methods for nonsmooth optimization, developed in [IK1] for the variational formulation of image restoration problems involving bounded variation type energy criterion. The Uzawa’s algorithm, first order augmented Lagrangian methods and Newton-like update using the active set strategy are described.

scholarly journals Stationary points for set-valued mappings on two metric spaces

2001 ◽  
Vol 28 (7) ◽  
pp. 427-432
Author(s):  
Zeqing Liu ◽  
Qingtao Liu ◽  
Shin Min Kang
Keyword(s):  
Stationary Point ◽  
Metric Spaces ◽  
Stationary Points ◽  
Compact Metric Spaces ◽  
Set Valued Mappings

We give stationary point theorems of set-valued mappings in complete and compact metric spaces. The results in this note generalize a few results due to Fisher.

Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC

2018 ◽  
Vol 35 (01) ◽  
pp. 1850008
Author(s):  
Na Xu ◽  
Xide Zhu ◽  
Li-Ping Pang ◽  
Jian Lv
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Accumulation Point ◽  
Stationary Points ◽  
Nondegeneracy Condition ◽  
Equilibrium Constraints ◽  
Convergence Properties ◽  
Constant Rank Constraint Qualification ◽  
Relaxation Schemes

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.

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