Calculus of convex polyhedra and polyhedral convex functions by utilizing a multiple objective linear programming solver

ABSTRACT

The article deals with operations defined on convex polyhedra or polyhedral convex functions. Given two convex polyhedra, operations like Minkowski sum, intersection and closed convex hull of the union are considered. Basic operations for one convex polyhedron are, for example, the polar, the conical hull and the image under affine transformation. The concept of a P-representation of a convex polyhedron is introduced. It is shown that many polyhedral calculus operations can be expressed explicitly in terms of P-representations. We point out that all the relevant computational effort for polyhedral calculus consists in computing projections of convex polyhedra. In order to compute projections we use a recent result saying that multiple objective linear programming (MOLP) is equivalent to the polyhedral projection problem. Based on the MOLP solver bensolve a polyhedral calculus toolbox for Matlab and GNU Octave is developed. Some numerical experiments are discussed.     

A convex-analysis perspective on disjunctive cuts

We treat with tools from convex analysis the general problem of cutting planes, separating a point from a (closed convex) set P. Crucial for this is the computation of extreme points in the so-called reverse polar set, introduced by E. Balas in 1979. In the polyhedral case, this enables the computation of cuts that define facets of P. We exhibit three (equivalent) optimization problems to compute such extreme points; one of them corresponds to selecting a specific normalization to generate cuts. We apply the above development to the case where P is (the closed convex hull of) a union, and more particularly a union of polyhedra (case of disjunctive cuts). We conclude with some considerations on the design of efficient cut generators. The paper also contains an appendix, reviewing some fundamental concepts of convex analysis. Supported by NSF grant DMII-0352885, ONR grant N00014-03-1-0188, INRIA grant ODW and IBM.     

Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces

In this paper, for the multiple-sets split feasibility problem, that is to find a point closest to a family of closed convex subsets in one space such that its image under a linear bounded mapping will be closest to another family of closed convex subsets in the image space, we study several iterative methods for finding a solution, which solves a certain variational inequality. We show that particular cases of our algorithms are some improvements for existing ones in literature. We also give two numerical examples for illustrating our algorithms.     

Partial Convexity

We study OC-convexity, which is defined by the intersection of conic semispaces of partial convexity. We investigate an optimization problem for OC-convex sets and prove a Krein--Milman type theorem for OC-convexity. The relationship between OC-convex and functionally convex sets is studied. Topological and numerical aspects, as well as separability properties are described. An upper estimate for the Carathéodory number for OC-convexity is found. On the other hand, it happens that the Helly and the Radon number for OC-convexity are infinite. We prove that the OC-convex hull of any finite set of points is the union of finitely many polyhedra.     

Range sets for weak efficiency in multiobjective linear programming and a parametric polytopes intersection problem

ABSTRACT

The aim of this paper is to obtain the range set for a given multiobjective linear programming problem and a weakly efficient solution. The range set is the set of all values of a parameter such that a given weakly efficient solution remains efficient when the objective coefficients vary in a given direction. The problem was originally formulated by Benson in 1985 and left to be solved. We formulate an algorithm for determining the range set, based on some hard optimization problems. Due to toughness of these optimization problems, we propose also lower and upper bound approximation techniques. In the second part, we focus on topological properties of the range set. In particular, we prove that a range set is formed by a finite union of intervals and we propose upper bounds on the number of intervals. Our approach to tackle the range set problem is via the intersection problem of parametric polytopes. Thus, our results have much wider area of applicability since the intersection (and separability) problem of convex polyhedra is important in many fields of optimization.     

Simplicial Branch-and-Reduce Algorithm for Convex Programs with a Multiplicative Constraint

Optimization problems that involve products of convex functions in the objective function or in the constraints arise in a variety of applications. These problems are difficult global optimization problems. During the past 15 years, however, a number of practical algorithms have been proposed for globally solving these types of problems. In this article, we present and validate a branch-and-reduce algorithm for finding a global optimal solution to a convex program that contains an additional constraint on the product of several convex functions. To globally solve this problem, the algorithm instead globally solves an equivalent master problem. At any stage of the algorithm, a disconnected set consisting of a union of simplices is constructed. This set is guaranteed to contain a portion of the boundary of the feasible region of the master problem where a global optimal solution lies. The algorithm uses a new branch-and-reduce scheme to iteratively reduce the sizes of these sets until a global optimal solution is found. Several potential computational advantages of the algorithm are explained, and a numerical example is solved.     

Almost sure convergence of random projected proximal and subgradient algorithms for distributed nonsmooth convex optimization

Two distributed algorithms are described that enable all users connected over a network to cooperatively solve the problem of minimizing the sum of all users’ objective functions over the intersection of all users’ constraint sets, where each user has its own private nonsmooth convex objective function and closed convex constraint set, which is the intersection of a number of simple, closed convex sets. One algorithm enables each user to adjust its estimate using the proximity operator of its objective function and the metric projection onto one constraint set randomly selected from a number of simple, closed convex sets. The other determines each user’s estimate using the subdifferential of its objective function instead of the proximity operator. Investigation of the two algorithms’ convergence properties for a diminishing step-size rule revealed that, under certain assumptions, the sequences of all users generated by each of the two algorithms converge almost surely to the same solution. It also showed that the rate of convergence depends on the step size and that a smaller step size results in quicker convergence. The results of numerical evaluation using a nonsmooth convex optimization problem support the convergence analysis and demonstrate the effectiveness of the two algorithms.     

A hierarchical method for real-time distance computation among moving convex bodies

This paper presents the Hierarchical Walk, or H-Walk algorithm, which maintains the distance between two moving convex bodies by exploiting both motion coherence and hierarchical representations. For convex polygons, we prove that H-Walk improves on the classic Lin–Canny and Dobkin–Kirkpatrick algorithms. We have implemented H-Walk for moving convex polyhedra in three dimensions. Experimental results indicate that, unlike previous incremental distance computation algorithms, H-Walk adapts well to variable coherence in the motion and provides consistent performance.     

On the convex hull of the union of certain polyhedra

We consider a finite collection of polyhedra whose defining linear systems differ only in their right hand sides. Jeroslow [5] and Blair [4] specified conditions under which the convex hull of the union of these polyhedra is defined by a system whose left hand side is the common lefthand side of the individual systems, and whose right hand side is a convex combination of the individual right hand sides. We give a new sufficient condition for this property to hold, which is often easier to recognize. In particular, we show that the condition is satisfied for polyhedra whose defining systems involve the node-arc incidence matrices of directed graphs, with certain righ hand sides. We also derive as a special case the compact linear characterization of the two terminal Steiner tree polytope given in Ball, Liu and Pulleyblank [3].     

On the range sets of variational inequalities

This paper investigates the closedness and convexity of the range sets of the variational inequality (VI) problem defined by an affine mappingM and a nonempty closed convex setK. It is proved that the range set is closed ifK is the union of a polyhedron and a compact convex set. Counterexamples are given such that the range set is not closed even ifK is a simple geometrical figure such as a circular cone or a circular cylinder in a three-dimensional space. Several sufficient conditions for closedness and convexity of the range set are presented. Characterization for the convex hull of the range set is established in the case whereK is a cone, while characterization for the closure of the convex hull of the range set is established in general. Finally, some applications to stability of VI problems are derived.This work was supported by the Australian Research Council.We are grateful to Professors M. Seetharama Gowda, Olvi Mangasarian, Jong-Shi Pang, and Steve Robinson for references. We are thankful to Professor Jim Burke for discussions on Theorem 2.1 and Counterexample 3.5.     

On Finite Linear Systems Containing Strict Inequalities

This paper deals with linear systems containing finitely many weak and/or strict inequalities, whose solution sets are referred to as evenly convex polyhedral sets. The classical Motzkin theorem states that every (closed and convex) polyhedron is the Minkowski sum of a convex hull of finitely many points and a finitely generated cone. In this sense, similar representations for evenly convex polyhedra have been recently given by using the standard version for classical polyhedra. In this work, we provide a new dual tool that completely characterizes finite linear systems containing strict inequalities and it constitutes the key for obtaining a generalization of Motzkin theorem for evenly convex polyhedra.     

Optimality Conditions in Terms of Alternance: Two Approaches

In Optimization Theory, necessary and sufficient optimality conditions play an essential role. They allow, first of all, checking whether a point under study satisfies the conditions, and, secondly, if it does not, finding a “better” point. For the class of directionally differentiable functions, a necessary condition for an unconstrained minimum requires the directional derivative to be non-negative in all directions. This condition becomes efficient for special classes of directionally differentiable functions. For example, in the case of convex and max-type functions, the necessary condition for a minimum takes the form of inclusion. The problem of verifying this condition is reduced to that of finding the point of some convex and compact set C which is nearest to the origin. If the origin does not belong to C, we easily find the steepest descent direction, and are able to construct a numerical method. In the classical Chebyshev polynomial approximation problem, necessary optimality conditions are expressed in the form of alternation of signs of some values. In the present paper, a generalization of the alternance approach to a general optimization problem is given. Two equivalent forms of the alternance condition (the so-called inside form and the outside one) are discussed in detail. In some cases, it may be more convenient to use the conditions in the form of inclusion, in some other—the condition in the alternance form as in the Chebyshev approximation problem. Numerical methods based on the condition in the form of inclusion usually are “gradient-type” methods, while methods employing the alternance form are often “Newton-type”. It is hoped that in some cases it will be possible to enrich the existing armory of optimization algorithms by a new family of efficient tools. In the paper, we discuss only unconstrained optimization problems in the finite-dimensional setting. In many cases, a constrained optimization problem can be reduced (via Exact Penalization Techniques) to an unconstrained one.     

Inaccurate linear equation system with a restricted-rank error matrix

It is well-known that the solution set of an interval linear equation system is a union of convex polyhedra the number of which increases, in general, exponentially with the problem size. As a consequence, the problem of finding the interval hull of the solution set is NP-hard as J. Rohn and V. Kreinovich proved in [13]. The purpose of this paper is to show that the solution set analysis can be simplified substantially provided the rank of the error matrix is restricted even if the assumption of interval character of data errors is replaced by a more general one. Especially, in the case of a rank-one error matrix we have to look into at most two convex subsets. Besides, a dual approach to describing the solution set is discussed. The original version of this approach was suggested in [7].     

The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces

In general normed spaces,we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior.We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function.Moreover,we provide necessary and suffcient conditions about the existence of weak(sharp) Pareto solutions.     

An inexact proximal point method for solving generalized fractional programs

In this paper, we present several new implementable methods for solving a generalized fractional program with convex data. They are Dinkelbach-type methods where a prox-regularization term is added to avoid the numerical difficulties arising when the solution of the problem is not unique. In these methods, at each iteration a regularized parametric problem is solved inexactly to obtain an approximation of the optimal value of the problem. Since the parametric problem is nonsmooth and convex, we propose to solve it by using a classical bundle method where the parameter is updated after each ‘serious step’. We mainly study two kinds of such steps, and we prove the convergence and the rate of convergence of each of the corresponding methods. Finally, we present some numerical experience to illustrate the behavior of the proposed algorithms, and we discuss the practical efficiency of each one.      

A method to convexify functions via curve evolution

This paper is devoted to a new method which allows to compute the convex envelope of a given function, by an evolution equation and techniques of curve evolution. We study the problem in the context of viscosity solutions and we propose numerical algorithms, to convexify a function, in one and two dimensions. In the end, we validate the model by presenting various numerical results. Key words: convex envelope, curve evolution, viscosity solutions, nonlinear second order elliptic and parabolic PDE's, finite differences scheme.     

Unique lifting of integer variables in minimal inequalities

This paper contributes to the theory of cutting planes for mixed integer linear programs (MILPs). Minimal valid inequalities are well understood for a relaxation of an MILP in tableau form where all the nonbasic variables are continuous; they are derived using the gauge function of maximal lattice-free convex sets. In this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities. We characterize precisely when the lifted coefficient is equal to the coefficient of the corresponding continuous variable in every minimal lifting (This result first appeared in the proceedings of IPCO 2010). The answer is a nonconvex region that can be obtained as a finite union of convex polyhedra. We then establish a necessary and sufficient condition for the uniqueness of the lifting function.     

Averaged Subgradient Methods for Constrained Convex Optimization and Nash Equilibria Computation

We consider the problem of finding minima of convex functions under convex inequality constraints as well as the problem of finding Nash equilibria in n -person constant sum games. We prove that both problems can be solved by algorithms whose basic principles consist of representing the original problems as infinite systems of convex inequalities which, in turn, can be approached by outer projection techniques. Experiments showing how one of these algorithms behaves in test cases are presented and, in context, we describe a numerical method for computing subgradients of convex functions.     

The interior proximal extragradient method for solving equilibrium problems

In this article we present a new and efficient method for solving equilibrium problems on polyhedra. The method is based on an interior-quadratic proximal term which replaces the usual quadratic proximal term. This leads to an interior proximal type algorithm. Each iteration consists in a prediction step followed by a correction step as in the extragradient method. In a first algorithm each of these steps is obtained by solving an unconstrained minimization problem, while in a second algorithm the correction step is replaced by an Armijo-backtracking linesearch followed by an hyperplane projection step. We prove that our algorithms are convergent under mild assumptions: pseudomonotonicity for the two algorithms and a Lipschitz property for the first one. Finally we present some numerical experiments to illustrate the behavior of the proposed algorithms.