Implicit function theorems and lagrange multipliers

Several implicit function theorems are proved, for systems of inequalities as well as equalities, assuming weaker differentiability than continuous Fréchet. These results give sufficient hypotheses for the Kuhn-Tucker conditions to hold, in a constrained minimization problem, with cone constraints and in any dimension. A condition is given for an inequality constraint system to be equivalently replaced by another, for which the function involved has surjective derivative. This allows some known implicit function theorems to be applied to inequality systems.     

On the Determination of a Density Function by Its Autoconvolution Coefficient

We investigate eigenvalues and eigenvectors of certain linear variational eigenvalue inequalities where the constraints are defined by a convex cone as in [4], [7], [8], [10]-[12], [17]. The eigenvalues of those eigenvalue problems are of interest in connection with bifurcation from the trivial solution of nonlinear variational inequalities. A rather far reaching theory is presented for the case that the cone is given by a finite number of linear inequalities, where an eigensolution corresponds to a (+)-Kuhn-Tucker point of the Rayleigh quotient. Application to an unlaterally supported beam are discussed and numerical results are given.     

Variational inequalities with cone constraints

The present paper considers variational inequalities with cone and operator constraints. The characterization of solutions is given. An algorithm for numerical solution of the problem is proposed, and the convergence of the generated sequence of points is proved.This research was supported in part by the Science Foundation of the Republic of Serbia.The author thanks two anonymous referees for their helpful remarks.     

Image space analysis for variational inequalities with cone constraints and applications to traffic equilibria

In this paper,the image space analysis (for short,ISA) is employed to investigate variational in- equalities (for short,VI) with cone constraints.Linear separation for VI with cone constraints is characterized by using the normal cone to a regularization of the image,and saddle points of the generalized Lagrangian func- tion.Lagrangian-type necessary and sufficient optimality conditions for VI with cone constraints are presented by using a separation theorem.Gap functions and weak sharpness for VI with cone constraints are also investi- gated.Finally,the obtained results are applied to standard and time-dependent traffic equilibria introduced by Daniele,Maugeri and Oettli.     

The Γ-algorithm and some applications

In this paper the power of the Γ-algorithm for obtaining the dual of a given cone and some of its multiple applications is discussed. The meaning of each sequential tableau appearing during the process is interpreted. It is shown that each tableau contains the generators of the dual cone of a given cone and that the algorithm updates the dual cone when new generators are incorporated. This algorithm, which is based on the duality concept, allows one to solve many problems in linear algebra, such as determining whether or not a vector belongs to a cone, obtaining the minimal representations of a cone in terms of a linear space and an acute cone, obtaining the intersection of two cones, discussing the compatibility of linear systems of inequalities, solving systems of linear inequalities, etc. The applications are illustrated with examples.     

Algorithm for computing the covering constant of a linear operator on a cone

An algorithm for computing the covering constant for the restriction of a linear operator to a cone defined by a finite set of inequalities is proposed. After a finite number of steps, the algorithm reduces the original problem to one of finding the eigenvalues of linear operators.     

An alogrithm for monotonic global optimization problems ∗

We propose an algorithm to locate a global maximum of an increasing function subject to an increasing constraint on the cone of vectors with nonnegative coordinates. The algorithm is based on the outer approximation of the feasible set. We eastablish the con vergence of the algorithm and provide a number of numerical experiments. We also discuss the types of constraints and objective functions for which the algorithm is best suited     

Lagrange Multipliers and mixed formulations for some inequality constrained variational inequalities and some nonsmooth unilateral problems

Abstract

This paper addresses a class of inequality constrained variational inequalities and nonsmooth unilateral variational problems. We present mixed formulations arising from Lagrange multipliers. First we treat in a reflexive Banach space setting the canonical case of a variational inequality that has as essential ingredients a bilinear form and a non-differentiable sublinear, hence convex functional and linear inequality constraints defined by a convex cone. We extend the famous Brezzi splitting theorem that originally covers saddle point problems with equality constraints, only, to these nonsmooth problems and obtain independent Lagrange multipliers in the subdifferential of the convex functional and in the ordering cone of the inequality constraints. For illustration of the theory we provide and investigate an example of a scalar nonsmooth boundary value problem that models frictional unilateral contact problems in linear elastostatics. Finally we discuss how this approach to mixed formulations can be further extended to variational problems with nonlinear operators and equilibrium problems, and moreover, to hemivariational inequalities.     

Tropical polar cones, hypergraph transversals, and mean payoff games

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.     

变分不等式与互补问题、双层规划与平衡约束数学规划问题的若干进展

考虑有限维变分不等式与互补问题、双层规划以及均衡约束的数学规划问题. 在简单介绍这些问题之后,重点介绍近年来这些领域中发展迅速的几个研究方向,包括对称锥互补问题的理论与算法、变分不等式的投影收缩算法、随机变分不等式与随机互补问题的模型与方法、双层规划以及均衡约束数学规划问题的新方法. 最后提出几个进一步研究的方向.     

An Adaptive Dual Parametrization Algorithm for Quadratic Semi-infinite Programming Problems

The so called dual parametrization method for quadratic semi-infinite programming (SIP) problems is developed recently for quadratic SIP problems with a single infinite constraint. A dual parametrization algorithm is also proposed for numerical solution of such problems. In this paper, we consider quadratic SIP problems with positive definite objective and multiple linear infinite constraints. All the infinite constraints are supposed to be continuously dependent on their index variable on a compact set which is defined by a number equality and inequalities. We prove that in the multiple infinite constraint case, the minimu parametrization number, just as in the single infinite constraint case, is less or equal to the dimension of the SIP problem. Furthermore, we propose an adaptive dual parametrization algorithm with convergence result. Compared with the previous dual parametrization algorithm, the adaptive algorithm solves subproblems with much smaller number of constraints. The efficiency of the new algorithm is shown by solving a number of numerical examples.     

New inequalities for the General Routing Problem

A large new class of valid inequalities is introduced for the General Routing Problem which properly contains the class of ‘K-C constraints’. These are also valid for the Rural Postman Problem. A separation algorithm is given for a subset of these inequalities which runs in polynomial time.     

An entropy-like proximal algorithm and the exponential multiplier method for convex symmetric cone programming

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we show that, under some mild assumptions, the sequence generated by the proposed algorithm is bounded and every accumulation point is a solution of the considered problem. In addition, we also present a dual application of the proposed algorithm to the symmetric cone linear program, leading to a multiplier method which is shown to possess similar properties as the exponential multiplier method (Tseng and Bertsekas in Math. Program. 60:1–19, 1993) holds.     

On Generalizations of the Frank-Wolfe Theorem to Convex and Quasi-Convex Programmes

In this paper we are concerned with the problem of boundedness and the existence of optimal solutions to the constrained optimization problem. We present necessary and sufficient conditions for boundedness of either a faithfully convex or a quasi-convex polynomial function over the feasible set defined by a system of faithfully convex inequality constraints and/or quasi-convex polynomial inequalities, where the faithfully convex functions satisfy some mild assumption. The conditions are provided in the form of an algorithm, terminating after a finite number of iterations, the implementation of which requires the identification of implicit equality constraints in a homogeneous linear system. We prove that the optimal solution set of the considered problem is nonempty, this way extending the attainability result well known as the so-called Frank-Wolfe theorem. Finally we show that our extension of the Frank-Wolfe theorem immediately implies continuity of the solution set defined by the considered system of (quasi)convex inequalities.     

Nonlinear Perturbations of Polyhedral Normal Cone Mappings and Affine Variational Inequalities

This paper establishes an upper estimate for the Fréchet normal cone to the graph of the nonlinearly perturbed polyhedral normal cone mappings in finite dimensional spaces. Under a positive linear independence assumption on the normal vectors of the active constraints at the point in question, the result leads to an upper estimate for values of the Mordukhovich coderivative of such mappings. On the basis, new results on solution stability of parametric affine variational inequalities under nonlinear perturbations are derived.     

On optimality conditions for vector variational inequalities

Optimality conditions for weak efficient, global efficient and efficient solutions of vector variational inequalities with constraints defined by equality, cone and set constraints are derived. Under various constraint qualifications, necessary optimality conditions for weak efficient, global efficient and efficient solutions in terms of the Clarke and Michel–Penot subdifferentials are established. With assumptions on quasiconvexity of constraint functions sufficient optimality conditions are also given.     

On Valid Inequalities for Mixed Integer p-Order Cone Programming

We discuss two families of valid inequalities for linear mixed integer programming problems with cone constraints of arbitrary order, which arise in the context of stochastic optimization with downside risk measures. In particular, we extend the results of Atamtürk and Narayanan (Math. Program., 122:1–20, 2010, Math. Program., 126:351–363, 2011), who developed mixed integer rounding cuts and lifted cuts for mixed integer programming problems with second-order cone constraints. Numerical experiments conducted on randomly generated problems and portfolio optimization problems with historical data demonstrate the effectiveness of the proposed methods.     

Second-Order Cone and Semidefinite Representations of Material Failure Criteria

A class of important problems in structural mechanics leads to optimization problems with linear objective functions and constraints consisting in (a) linear equalities and (b) inequalities imposed by the material strength, the so-called failure criteria. It is shown that a wide variety of failure criteria can be represented as systems of either second-order cone or semidefinite constraints, giving rise to respective optimization problems. Work partially supported by Air Force grants.     

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.     

Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation

We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP). We first develop a general conic program relaxation for QCQP based on a matrix decomposition scheme and polyhedral (piecewise linear) underestimation. By employing suitable matrix cones, we then show that the convex conic relaxation can be reduced to a semidefinite programming (SDP) problem. In particular, we investigate polyhedral underestimations for several classes of matrix cones, including the cones of rank-1 and rank-2 matrices, the cone generated by the coefficient matrices, the cone of positive semidefinite matrices and the cones induced by rank-2 semidefinite inequalities. We demonstrate that in general the new SDP relaxations can generate lower bounds at least as tight as the best known SDP relaxations for QCQP. Moreover, we give examples for which tighter lower bounds can be generated by the new SDP relaxations. We also report comparison results of different convex relaxation schemes for nonconvex QCQP with convex quadratic/linear constraints, nonconvex quadratic constraints and 0–1 constraints.