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  • 1
    Electronic Resource
    Electronic Resource
    Multiplicative Programming Problems: Analysis and Efficient Point Search Heuristic (1997)
    Springer
    Journal of optimization theory and applications 94 (1997), S. 487-510 
    Details
    ISSN: 1573-2878
    Keywords: Multiplicative programming ; global optimization ; concave minimization ; efficient points ; heuristic algorithms ; multiple objectives
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Multiplicative programming problems are difficult global optimization problems known to be NP-hard. At the same time, these problems have some important applications in engineering, finance, economics, and other fields. This article has two purposes. The first is to present an analysis that shows several relationships between concave multiplicative programs and concave minimization problems, and between concave multiplicative programs and certain multiple-objective mathematical programs. The second purpose is to propose and report computational results for a heuristic efficient-point search algorithm that we have designed for use on linear multiplicative programming problems. To our knowledge, this is the first heuristic algorithm of its type. The theoretical and algorithmic results given in the article offer some potentially important new avenues for analyzing and solving multiplicative programming problems of various types.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022600232285
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  • 2
    Electronic Resource
    Electronic Resource
    Optimization over the efficient set: Four special cases (1994)
    Springer
    Journal of optimization theory and applications 80 (1994), S. 3-18 
    Details
    ISSN: 1573-2878
    Keywords: Multiple criteria decision making ; efficient set ; global optimization ; linear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Recently, researchers and practitioners have been increasingly interested in the problem (P) of maximizing a linear function over the efficient set of a multiple objective linear program. Problem (P) is generally a difficult global optimization problem which requires numerically intensive procedures for its solution. In this paper, simple linear programming procedures are described for detecting and solving four special cases of problem (P). When solving instances of problem (P), these procedures can be used as screening devices to detect and solve these four special cases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02196590
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  • 3
    Electronic Resource
    Electronic Resource
    Outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem (1996)
    Springer
    Journal of optimization theory and applications 88 (1996), S. 77-105 
    Details
    ISSN: 1573-2878
    Keywords: Multiple criteria decision making ; efficient set ; global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This article presents a finite, outcome-based algorithm for optimizing a lower semicontinuous function over the efficient set of a bicriteria linear programming problem. The algorithm searches the efficient faces of the outcome set of the bicriteria linear programming problem. It exploits the fact that the dimension of the outcome set of the bicriteria problem is at most two. As a result, in comparison to decisionbased approaches, the number of efficient faces that need to be found is markedly reduced. Furthermore, the dimensions of the efficient faces found are always at most one. The algorithm can be implemented for a wide variety of lower semicontinuous objective functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02192023
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  • 4
    Electronic Resource
    Electronic Resource
    Hybrid Approach for Solving Multiple-Objective Linear Programs in Outcome Space (1998)
    Springer
    Journal of optimization theory and applications 98 (1998), S. 17-35 
    Details
    ISSN: 1573-2878
    Keywords: Multiple-objective linear programming ; vector maximization ; efficient set ; outcome set ; global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Various difficulties arise in using decision set-based vector maximization methods to solve a multiple-objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have begun to develop tools for analyzing and solving problem (MOLP) in outcome space, rather than in decision space. In this article, we present and validate a new hybrid vector maximization approach for solving problem (MOLP) in outcome space. The approach systematically integrates a simplicial partitioning technique into an outer approximation procedure to yield an algorithm that generates the set of all efficient extreme points in the outcome set of problem (MOLP) in a finite number of iterations. Some key potential practical and computational advantages of the approach are indicated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022628612489
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  • 5
    Electronic Resource
    Electronic Resource
    Generalized γ-Valid Cut Procedure for Concave Minimization (1999)
    Springer
    Journal of optimization theory and applications 102 (1999), S. 289-298 
    Details
    ISSN: 1573-2878
    Keywords: Concave minimization ; γ-valid cut ; Tuy cut ; global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The concept of a γ-valid cutting plane has been used in many types of algorithms for solving concave minimization problems. Unfortunately, the procedures proposed to date for constructing these cuts are valid only under certain assumptions that often may not hold in practice. Chief among these is the requirement that the feasible region of the concave minimization problem in question have full dimension, and that the objective function of this problem be concave rather than quasiconcave. In this article, we propose, validate, and show how to implement a more general γ-valid cutting plane procedure which eliminates these restrictions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021776323080
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  • 6
    Electronic Resource
    Electronic Resource
    Outcome-Space Cutting-Plane Algorithm for Linear Multiplicative Programming (2000)
    Springer
    Journal of optimization theory and applications 104 (2000), S. 301-322 
    Details
    ISSN: 1573-2878
    Keywords: global optimization ; multiplicative programming ; cutting plane ; nonconvex programming ; outcome space ; extreme-point search
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This article presents an outcome-space pure cutting-plane algorithm for globally solving the linear multiplicative programming problem. The framework of the algorithm is taken from a pure cutting-plane decision set-based method developed by Horst and Tuy for solving concave minimization problems. By adapting this method to an outcome-space reformulation of the linear multiplicative programming problem, rather than applying directly the method to the original decision-set formulation, it is expected that considerable computational savings can be obtained. Also, we show how additional computational benefits might be obtained by implementing the new algorithm appropriately. To illustrate the new algorithm, we apply it to the solution of a sample problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1004657629105
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  • 7
    Electronic Resource
    Electronic Resource
    A finite, nonadjacent extreme-point search algorithm for optimization over the efficient set (1992)
    Springer
    Journal of optimization theory and applications 73 (1992), S. 47-64 
    Details
    ISSN: 1573-2878
    Keywords: Multiple-criteria decision making ; extreme-point search ; global optimization ; efficient set ; nonconvex programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The problem (P) of optimizing a linear function over the efficient set of a multiple-objective linear program serves many useful purposes in multiple-criteria decision making. Mathematically, problem (P) can be classified as a global optimization problem. Such problems are much more difficult to solve than convex programming problems. In this paper, a nonadjacent extreme-point search algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact extreme-point optimal solution for the problem after a finite number of iterations. It can be implemented using only linear programming methods. Convergence of the algorithm is proven, and a discussion is included of its main advantages and disadvantages.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940077
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