Showing papers in "Journal of Global Optimization in 1995"

Greedy Randomized Adaptive Search Procedures

Abstract: Today, a variety of heuristic approaches are available to the operations research practitioner. One methodology that has a strong intuitive appeal, a prominent empirical track record, and is trivial to efficiently implement on parallel processors is GRASP (Greedy Randomized Adaptive Search Procedures). GRASP is an iterative randomized sampling technique in which each iteration provides a solution to the problem at hand. The incumbent solution over all GRASP iterations is kept as the final result. There are two phases within each GRASP iteration: the first intelligently constructs an initial solution via an adaptive randomized greedy function; the second applies a local search procedure to the constructed solution in hope of finding an improvement. In this paper, we define the various components comprising a GRASP and demonstrate, step by step, how to develop such heuristics for combinatorial optimization problems. Intuitive justifications for the observed empirical behavior of the methodology are discussed. The paper concludes with a brief literature review of GRASP implementations and mentions two industrial applications.

αBB: A global optimization method for general constrained nonconvex problems

Abstract: A branch and bound global optimization method,αBB, for general continuous optimization problems involving nonconvexities in the objective function and/or constraints is presented. The nonconvexities are categorized as being either of special structure or generic. A convex relaxation of the original nonconvex problem is obtained by (i) replacing all nonconvex terms of special structure (i.e. bilinear, fractional, signomial) with customized tight convex lower bounding functions and (ii) by utilizing the α parameter as defined in [17] to underestimate nonconvex terms of generic structure. The proposed branch and bound type algorithm attains finitee-convergence to the global minimum through the successive subdivision of the original region and the subsequent solution of a series of nonlinear convex minimization problems. The global optimization method,αBB, is implemented in C and tested on a variety of example problems.

Bilevel programming in traffic planning: Models, methods and challenge

Abstract: Well-founded traffic models recognize the individual network user's right to the decision as to when, where and how to travel. On the other hand, the decisions concerning management, control, design and improvement investments are made by the public sector in the interest of the society as a whole. Hence, transportation planning is a characteristic example of a hierarchical process, in which the public sector at one level makes decisions seeking to improve the performance of the network, while at another level the network users make choices with regard to route, travel mode, origin and destination of their travel. Our objective is to provide a review on the current state of research and development in bilevel programming problems that arize in this context, and attract the attention of the global optimization community to this problem class of imense practical importance.

Finding all solutions of nonlinearly constrained systems of equations

Abstract: A new approach is proposed for finding alle-feasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem (P) whose multiple global minimum solutions with a zero objective value (if any) correspond to all solutions of the initial constrained system of equalities. Alle-globally optimal points of (P) are then localized within a set of arbitrarily small disjoint rectangles. This is based on a branch and bound type global optimization algorithm which attains finitee-convergence to each of the multiple global minima of (P) through the successive refinement of a convex relaxation of the feasible region and the subsequent solution of a series of nonlinear convex optimization problems. Based on the form of the participating functions, a number of techniques for constructing this convex relaxation are proposed. By taking advantage of the properties of products of univariate functions, customized convex lower bounding functions are introduced for a large number of expressions that are or can be transformed into products of univariate functions. Alternative convex relaxation procedures involve either the difference of two convex functions employed in αBB [23] or the exponential variable transformation based underestimators employed for generalized geometric programming problems [24]. The proposed approach is illustrated with several test problems. For some of these problems additional solutions are identified that existing methods failed to locate.

A recipe for semidefinite relaxation for (0,1)-quadratic programming: In memory of Svata Poljak

Abstract: We review various relaxations of (0,1)-quadratic programming problems. These include semidefinite programs, parametric trust region problems and concave quadratic maximization. All relaxations that we consider lead to efficiently solvable problems. The main contributions of the paper are the following. Using Lagrangian duality, we prove equivalence of the relaxations in a unified and simple way. Some of these equivalences have been known previously, but our approach leads to short and transparent proofs. Moreover we extend the approach to the case of equality constrained problems by taking the squared linear constraints into the objective function. We show how this technique can be applied to the Quadratic Assignment Problem, the Graph Partition Problem and the Max-Clique Problem. Finally we show our relaxation to be best possible among all quadratic majorants with zero trace.

Some geometric results in semidefinite programming

Abstract: The purpose of this paper is to develop certain geometric results concerning the feasible regions of Semidefinite Programs, called hereSpectrahedra. We first develop a characterization for the faces of spectrahedra. More specifically, given a pointx in a spectrahedron, we derive an expression for the minimal face containingx. Among other things, this is shown to yield characterizations for extreme points and extreme rays of spectrahedra. We then introduce the notion of an algebraic polar of a spectrahedron, and present its relation to the usual geometric polar.

A reformulation-convexification approach for solving nonconvex quadratic programming problems

Abstract: In this paper, we consider the class of linearly constrained nonconvex quadratic programming problems, and present a new approach based on a novel Reformulation-Linearization/Convexification Technique. In this approach, a tight linear (or convex) programming relaxation, or outer-approximation to the convex envelope of the objective function over the constrained region, is constructed for the problem by generating new constraints through the process of employing suitable products of constraints and using variable redefinitions. Various such relaxations are considered and analyzed, including ones that retain some useful nonlinear relationships. Efficient solution techniques are then explored for solving these relaxations in order to derive lower and upper bounds on the problem, and appropriate branching/partitioning strategies are used in concert with these bounding techniques to derive a convergent algorithm. Computational results are presented on a set of test problems from the literature to demonstrate the efficiency of the approach. (One of these test problems had not previously been solved to optimality). It is shown that for many problems, the initial relaxation itself produces an optimal solution.

A global optimization algorithm for linear fractional and bilinear programs

Abstract: In this paper a deterministic method is proposed for the global optimization of mathematical programs that involve the sum of linear fractional and/or bilinear terms. Linear and nonlinear convex estimator functions are developed for the linear fractional and bilinear terms. Conditions under which these functions are nonredundant are established. It is shown that additional estimators can be obtained through projections of the feasible region that can also be incorporated in a convex nonlinear underestimator problem for predicting lower bounds for the global optimum. The proposed algorithm consists of a spatial branch and bound search for which several branching rules are discussed. Illustrative examples and computational results are presented to demonstrate the efficiency of the proposed algorithm.

Global optimization for the Biaffine Matrix Inequality problem

Abstract: It has recently been shown that an extremely wide array of robust controller design problems may be reduced to the problem of finding a feasible point under a Biaffine Matrix Inequality (BMI) constraint. The BMI feasibility problem is the bilinear version of the Linear (Affine) Matrix Inequality (LMI) feasibility problem, and may also be viewed as a bilinear extension to the Semidefinite Programming (SDP) problem. The BMI problem may be approached as a biconvex global optimization problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices. This paper presents a branch and bound global optimization algorithm for the BMI. A simple numerical example is included. The robust control problem, i.e., the synthesis of a controller for a dynamic physical system which guarantees stability and performance in the face of significant modelling error and worst-case disturbance inputs, is frequently encountered in a variety of complex engineering applications including the design of aircraft, satellites, chemical plants, and other precision positioning and tracking systems.

On the selection of subdivision directions in interval branch-and-bound methods for global optimization

Abstract: This paper investigates the influence of the interval subdivision selection rule on the convergence of interval branch-and-bound algorithms for global optimization. For the class of rules that allows convergence, we study the effects of the rules on a model algorithm with special list ordering. Four different rules are investigated in theory and in practice. A wide spectrum of test problems is used for numerical tests indicating that there are substantial differences between the rules with respect to the required CPU time, the number of function and derivative evaluations, and the necessary storage space. Two rules can provide considerable improvements in efficiency for our model algorithm.

A relaxation method for nonconvex quadratically constrained quadratic programs

Abstract: We present an algorithm for finding approximate global solutions to quadratically constrained quadratic programming problems. The method is based on outer approximation (linearization) and branch and bound with linear programming subproblems. When the feasible set is non-convex, the infinite process can be terminated with an approximate (possibly infeasible) optimal solution. We provide error bounds that can be used to ensure stopping within a prespecified feasibility tolerance. A numerical example illustrates the procedure. Computational experiments with an implementation of the procedure are reported on bilinearly constrained test problems with up to sixteen decision variables and eight constraints.

Optimization over the efficient set

Abstract: This paper deals with the problem of maximizing a function over the efficient set of a linear multiple objective program. The approach is to formulate a biobjective program with an appropriate efficient set. The penalty function approach is motivated by an auxiliary problem due to Benson.

An Adaptive Simulated Annealing Algorithm for Global Optimization over Continuous Variables

Abstract: A method is presented for attempting global minimization for a function of continuous variables subject to constraints. The method, calledAdaptive Simulated Annealing (ASA), is distinguished by the fact that the fixed temperature schedules and step generation routines that characterize other implementations are here replaced by heuristic-based methods that effectively eliminate the dependence of the algorithm's overall performance on user-specified control parameters. A parallelprocessing version of ASA that gives increased efficiency is presented and applied to two standard problems for illustration and comparison.

The linear complementarity problem as a separable bilinear program

Abstract: The nonmonotone linear complementarity problem (LCP) is formulated as a bilinear program with separable constraints and an objective function that minimizes a natural error residual for the LCP. A linear-programming-based algorithm applied to the bilinear program terminates in a finite number of steps at a solution or stationary point of the problem. The bilinear algorithm solved 80 consecutive cases of the LCP formulation of the knapsack feasibility problem ranging in size between 10 and 3000, with almost constant average number of major iterations equal to four.

Integral global minimization: Algorithms, implementations and numerical tests

Abstract: The theoretical foundation of integral global optimization has become widely known and well accepted [4],[24],[25]. However, more effort is needed to demonstrate the effectiveness of the integral global optimization algorithms. In this work we detail the implementation of the integral global minimization algorithms. We describe how the integral global optimization method handles nonconvex unconstrained or box constrained, constrained or discrete minimization problems. We illustrate the flexibility and the efficiency of integral global optimization method by presenting the performance of algorithms on a collection of well known test problems in global optimization literature. We provide the software which solves these test problems and other minimization problems. The performance of the computations demonstrates that the integral global algorithms are not only extremely flexible and reliable but also very efficient.

A D.C. optimization method for single facility location problems

Abstract: The single facility location problem with general attraction and repulsion functions is considered. An algorithm based on a representation of the objective function as the difference of two convex (d.c.) functions is proposed. Convergence to a global solution of the problem is proven and extensive computational experience with an implementation of the procedure is reported for up to 100,000 points. The procedure is also extended to solve conditional and limited distance location problems. We report on limited computational experiments on these extensions.

Computing global minima to polynomial optimization problems using Gröbner bases

Abstract: The local optimality conditions to polynomial optimization problems are a set of polynomial equations (plus some inequality conditions). With the recent techniques of Grobner bases one can find all solutions to such systems, and hence also find global optima. We give a short survey of these methods. We also apply them to a set of problems termed ‘with exact solutions unknown’ in the problem sets of Hock and Schittkowski. To these problems we give exact solutions.

Solution of linear complementarity problems using minimization with simple bounds

Abstract: We define a minimization problem with simple bounds associated to the horizontal linear complementarity problem (HLCP). When the HLCP is solvable, its solutions are the global minimizers of the associated problem. When the HLCP is feasible, we are able to prove a number of properties of the stationary points of the associated problem. In many cases, the stationary points are solutions of the HLCP. The theoretical results allow us to conjecture that local methods for box constrained optimization applied to the associated problem are efficient tools for solving linear complementarity problems. Numerical experiments seem to confirm this conjecture.

A geometrical analysis of the efficient outcome set in multiple objective convex programs with linear criterion functions

Abstract: This article performs a geometrical analysis of the efficient outcome setY E of a multiple objective convex program (MLC) with linear criterion functions. The analysis elucidates the facial structure ofY E and of its pre-image, the efficient decision setX E . The results show thatY E often has a significantly-simpler structure thanX E . For instance, although both sets are generally nonconvex and their maximal efficient faces are always in one-to-one correspondence, large numbers of extreme points and faces inX E can map into non-facial subsets of faces inY E , but not vice versa. Simple tests for the efficiency of faces in the decision and outcome sets are derived, and certain types of faces in the decision set are studied that are immune to a common phenomenon called “collapsing”. The results seem to indicate that significant computational benefits may potentially be derived if algorithms for problem (MLC) were to work directly with the outcome set of the problem to find points and faces ofY E , rather than with the decision set.

On numerical solution of hemivariational inequalities by nonsmooth optimization methods

Abstract: In this paper we consider numerical solution of hemivariational inequalities (HVI) by using nonsmooth, nonconvex optimization methods. First we introduce a finite element approximation of (HVI) and show that it can be transformed to a problem of finding a substationary point of the corresponding potential function. Then we introduce a proximal budle method for nonsmooth nonconvex and constrained optimization. Numerical results of a nonmonotone contact problem obtained by the developed methods are also presented.

The Minimum Concave Cost Network Flow Problem with fixed numbers of sources and nonlinear arc costs

Abstract: We prove that the Minimum Concave Cost Network Flow Problem with fixed numbers of sources and nonlinear arc costs can be solved by an algorithm requiring a number of elementary operations and a number of evaluations of the nonlinear cost functions which are both bounded by polynomials inr, n, m, wherer is the number of nodes,n is the number of arcs andm the number of sinks in the network.

A branch and bound algorithm for bound constrained optimization problems without derivatives

Abstract: In this paper, we give a new branch and bound algorithm for the global optimization problem with bound constraints. The algorithm is based on the use of inclusion functions. The bounds calculated for the global minimum value are proved to be correct, all rounding errors are rigorously estimated. Our scheme attempts to exclude most “uninteresting” parts of the search domain and concentrates on its “promising” subsets. This is done as fast as possible (by involving local descent methods), and uses little information as possible (no derivatives are required). Numerical results for many well-known problems as well as some comparisons with other methods are given.

An algorithm for solving global optimization problems with nonlinear constraints

Abstract: In this paper we propose an algorithm using only the values of the objective function and constraints for solving one-dimensional global optimization problems where both the objective function and constraints are Lipschitzean and nonlinear. The constrained problem is reduced to an unconstrained one by the index scheme. To solve the reduced problem a new method with local tuning on the behavior of the objective function and constraints over different sectors of the search region is proposed. Sufficient conditions of global convergence are established. We also present results of some numerical experiments.

A method for solving d.c. programming problems. Application to fuel mixture nonconvex optimization problem

Abstract: We present a numerical method for solving the d.c. programming problem $$c^* = \min \{ \langle c,x\rangle s.t. f_i (x) \leqslant 0, i = 1,...,m, x \in D\} $$ wherefi, i=1,...,m are d.c. (difference of two convex functions) and D is a convex set in ℝn. An (ɛ, η)-solutionx(ɛ, η) satisfying $$x(\varepsilon ,\eta ) \in D, \langle c,x(\varepsilon ,\eta )\rangle \leqslant c^* + \varepsilon , f_i (x(\varepsilon ,\eta )) \leqslant \eta , i = 1,...,m,$$ can be found after a finite number of iterations. This algorithm combines an outer approximation procedure for solving a system of d.c. inequalities with a simple general scheme for minimizing a linear function over a compact set. As an application we discuss the numerical solution of a fuel mixture problem (encountered in the oil industry).

Planar point-objective location problems with nonconvex constraints: A geometrical construction

Abstract: The planar point-objective location problem has attracted considerable interest among Location Theory researchers. The result has been a number of papers giving properties or algorithms for particular instances of the problem. However, most of these results are only valid when the feasible region where the facility is to be located is the whole space ℝ2, which is a rather inaccurate approximation in many real world location problems.

On global maximum of a convex terminal functional in optimal control problems

Abstract: In this paper necessary and sufficient conditions (related to Pontryagin's principle) for a global maximum of a convex terminal functional for different types of control systems are proved. A few examples are given.

Nonconvex energy functions, related eigenvalue hemivariational inequalities on the sphere and applications

Abstract: The present paper deals with a new type of eigenvalue problems arising in problems involving nonconvex nonsmooth energy functions. They lead to the search of critical points (e.g. local minima) for nonconvex nonsmooth potential functions which in turn give rise to hemivariational inequalities. For this type of variational expressions the eigenvalue problem is studied here concerning the existence and multiplicity of solutions by applying a critical point theory appropriate for nonsmooth nonconvex functionals.

Semicoercive variational hemivariational inequalities

Abstract: The aim of this paper is the mathematical study of a general class of semicoercive variational hemivariational inequalities introduced by P.D. Panagiotopoulos in order to formulate problems of mechanics involving nonconvex and nonsmooth energy function. Our approach is based on the asymptotic behavior of the functions which are involved in the variational problems.

On complexity of subset interconnection designs

Abstract: Given a setX and subsetsX1,...,Xm, we consider the problem of finding a graphG with vertex setX and the minimum number of edges such that fori=1,...,m, the subgraphGi; induced byXi is connected. Suppose that for anyα pointsx1,...,xαe X, there are at mostβXi 's containing the set {x1,...,xα}. In the paper, we show that the problem is polynomial-time solvable for (α ⩽ 2,β ⩽ 2) and is NP-hard for (α⩾3,β=1), (α=l,β⩾6), and (α⩾2,β⩾3).