Malgorzata M. Wiecek

Routing with nonlinear multiattribute cost functions

Abstract The paper first points out the connection between the problem of finding a set of Pareto-optimal paths in the presence of multiple criteria and the problem of finding the optimal path in the presence of a single multiattribute criterion. A survey of literature indicates that the former problem has received much attention and several exact (but nonpolynomial) and approximate algorithms are available for finding all and nearly all Pareto-optimal paths, respectively. The latter problem of finding the optimal path that minimizes a nonlinear cost function of multiple attributes has received less attention. The paper examines the properties of the optimal path when the cost function is monotonic and concave in the attributes, especially how it relates to the set of “efficient” paths within the nondominated set. When the cost function in convex in two attributes, a line-search algorithm is developed that finds a good, if not optimal, path without using any assumptions or information on the derivatives of the cost function.

Linear complementarity problems and multiple objective programming

An equivalence is demonstrated between solving a linear complementarity problem with general data and finding a certain subset of the efficient points of a multiple objective programming problem. A new multiple objective programming based approach to solving linear complementarity problems is presented. Results on existence, uniqueness and computational complexity are included.

Vector optimization and generalized Lagrangian duality

In this paper, foundations of a new approach for solving vector optimization problems are introduced. Generalized Lagrangian duality, related for the first time with vector optimization, provides new scalarization techniques and allows for the generation of efficient solutions for problems which are not required to satisfy any convexity assumptions.

Multiple-objective programming with polynomial objectives and constraints

Abstract An approach to solving multiple-objective programming problems with polynomial objectives and polynomial constraints is developed. The notions of pre-efficient solution and local efficient solution are investigated relative to the single-objective problem of Benson. Illustrative examples are presented in which the solutions are obtained by means of homotopy continuation.

A Parallel Algorithm for Multiple Objective Linear Programs

This paper presents an ADBASE-based parallel algorithm forsolving multiple objective linear programs (MOLPs). Job balance,speedup and scalability are of primary interest in evaluatingefficiency of the new algorithm. The scalability of a parallelalgorithm is a measure of its capacity to increase performance withrespect to the number of processors used. Implementation results onIntel iPSC/2 and Paragon multiprocessors show that the algorithmsignificantly speeds up the process of solving MOLPs, which isunderstood as generating all or some efficient extreme points andunbounded efficient edges. The algorithm is shown to be scalable andgives better results for large problems. Motivation andjustification for solving large MOLPs are also included.

Efficiency and Solution Approaches to Bicriteria Nonconvex Programs

A new nonlinear scalarization specially designed for bicriteria nonconvexprogramming problems is presented. The scalarization is based on generalizedLagrangian duality theory and uses an augmented Lagrange function. The newconcepts, qi-approachable points and augmented duality gap, are introducedin order to determine the location of nondominated solutions with respect to aduality gap as well as the connectedness of the nondominated set.

An Augmented Lagrangian Scalarization for Multiple Objective Programming

The multiple objective program (MOP) is related to its e-constraint single-objective counterpart for which an augmented Lagrangian is developed. The resulting scalarization generates qk-approachable points that are identified as locally efficient and/or efficient points of the MOP. An illustrative example is enclosed.

A Flexible Approach to Piecewise Linear Multiple Objective Programming

An approach to generating all efficient solutions of multiple objective programs with piece- wise linear objective functions and linear constraints is presented. The approach is based on the decomposition of the feasible set into subsets, referred to as cells, so that the original problem reduces to a series of linear multiple objective programs over the cells. The concepts of cell-efficiency and complex-efficiency are introduced and their relationship with efficiency is examined. Applications in location theory as well as in worst case analysis are highlighted.

A Tchebycheff Metric Approach to the Optimal Path Problem with Nonlinear Multiattribute Cost Functions

A global optimization problem of finding an optimal path in the network with multiple attributes on links and a nonlinear convex cost function is studied. It is shown that the modified weighted Tchebycheff metric scalarization can generate every nondominated path in the network. Two exact algorithms for solving the bi-attribute optimal path problem are presented and an illustrative example is enclosed.

A bookkeeping strategy for multiple objective linear programs

This paper discusses the bookkeeping strategies for solving large multiple objective linear programs (MOLPs) on ADBASE, a well developed sequential software package, and on a parallel ADBASE algorithm. Three representative list creation schemes were first analyzed and tested. The best of them, Binary Search with Insertion Sort (BSIS), was selected to be incorporated into ADBASE and the parallel ADBASE algorithm. The resulting new bookkeeping strategy was then tested in ADBASE as well as implemented in the parallel ADBASE algorithm. The parallel implementations were carried out on an Intel Paragon multiprocessor. Computational results show that the new bookkeeping strategy for maintaining a list of efficient solutions significantly speeds up the process of solving MOLPs, especially on parallel computers.