Surrogate Multipliers Research Articles

On a New Constraint Reduction Heuristic Using Improved Bisection Method for Mixed Integer Linear Programming

In this study, we develop a surrogate relaxation-based procedure to reduce mixed-integer linear programming (MILP) problem sizes. This technique starts with one surrogate constraint which is a nonnegative linear combination of multiple constraints of the problem. At this initial step, we calculate optimal Lagrangian multipliers from LP relaxation of the problem and use them as initial surrogate multipliers. We incorporate the improved bisection method (IBM) (B. Gavish, F. Glover, and H. Pirkul, Surrogate Constraints in Integer Programming, J. Inform. Optim. Sci. 12(2) (1991), 219-228.) into our algorithm. This simple heuristic algorithm is designed to iteratively generate a new surrogate cut that is to guarantee to satisfy the most violated two constraints of the corresponding iteration. The performance of the heuristic is tested using both some problems from the OR libraries and randomly generated ones.

Surrogate upper bound sets for bi-objective bi-dimensional binary knapsack problems

The paper deals with the definition and the computation of surrogate upper bound sets for the bi-objective bi-dimensional binary knapsack problem. It introduces the Optimal Convex Surrogate Upper Bound set, which is the tightest possible definition based on the convex relaxation of the surrogate relaxation. Two exact algorithms are proposed: an enumerative algorithm and its improved version. This second algorithm results from an accurate analysis of the surrogate multipliers and the dominance relations between bound sets. Based on the improved exact algorithm, an approximated version is derived. The proposed algorithms are benchmarked using a dataset composed of three groups of numerical instances. The performances are assessed thanks to a comparative analysis where exact algorithms are compared between them, the approximated algorithm is confronted to an algorithm introduced in a recent research work.

On zero duality gap in surrogate constraint optimization: The case of rational-valued functions of constraints

This paper is concerned with the constrained optimization problem. A detailed discussion of surrogate constraints with zero duality gaps is presented. Readily available surrogate multipliers are considered that close the duality gaps where constraints are rational-valued. Through illustrative examples, the sources of duality gaps are examined in detail. While in the published literature, in many situations conclusions have been made about the existence of non-zero duality gaps, we show that taking advantage of full problem information can close the duality gaps. Overlooking such information can produce shortcomings in the research in which a non-zero duality gap is observed. We propose theorems to address the shortcomings and report results regarding implementation issues.

Surrogate relaxation of a fuzzy multidimensional 0–1 knapsack model by surrogate constraint normalization rules and a methodology for multi-attribute project portfolio selection

In this paper, a multidimensional 0–1 knapsack model with fuzzy parameters is defuzzified using triangular norm (t-norm) and t-conorm fuzzy relations. In the first part of the paper, the surrogate relaxation models of the defuzzified models are developed, and the use of surrogate constraint normalization rules is proposed as the surrogate multipliers. A methodology is proposed to evaluate some surrogate constraint normalization rules proposed in the literature as well as one rule proposed in this paper. Three distance metrics are used to find the distance of fuzzy objective function from the surrogate models to the distance of fuzzy objective function from the original models. A numerical experiment shows that the rule proposed in this paper dominates the other rules considered in this paper for three distance metrics given the whole assumptions. In the second part of the paper, a methodology is proposed for multi-attribute project portfolio selection, and optimal solutions from the original defuzzified models as well as near-optimal solutions from their surrogate relaxation models are considered as alternatives. The aggregation of evaluation results is managed using a simple yet effective method so-called fuzzy Simple Additive Weighting (SAW) method. Then, the methodology is applied to a hypothetical construction project portfolio selection problem with multiple attributes.

An approach for solving nonlinear multi-objective separable discrete optimization problem with one constraint

We propose an exact solution approach for solving nonlinear multi-objective optimization problems with separable discrete variables and a single constraint. The approach converts the multi-objective problem into a single objective problem by using surrogate multipliers from which we find all the solutions with objective values within a given range. We call this the surrogate target problem which is solved by using an algorithm based on the modular approach. Computational experiments demonstrate the effectiveness of this approach in solving large-scale problems. A simple example is presented to illustrate an interactive decision making process.

A method for determining optimal multipliers in the surrogate constraint method

It has recently been important to obtain optimal solutions to multidimensional nonlinear integer programming problems with multiple constraints. The surrogate constraint method is very effective in obtaining high-quality solutions of the original multidimensional problems. The surrogate constraint method translates a multidimensional problem into a one-dimensional problem by using a surrogate multiplier. But when there exists a surrogate duality gap between the translated one-dimensional problem and the original multidimensional problem, the optimal solution to the surrogate problem is not optimal with respect to the original problem. Nakagawa has recently proposed an improved surrogate constraint (ISC) method that can reduce the surrogate duality gap and provide an exact solution to the original problem. By using this method, we can obtain exact solutions to separable nonlinear integer programming problems with 1000 variables and 5 or 6 constraints. The ISC method requires an optimal surrogate multiplier. However, there are problems of calculation time and memory requirements in the algorithm for obtaining the optimal surrogate multiplier when the number of constraints is large. In this paper, the Dyer algorithm and Nakagawa's Cutting-Off Polyhedron (COP) algorithm for an algorithm that obtains the optimal surrogate multiplier in the ISC method are compared in computational experiments, and the advantages and disadvantages of these algorithms are investigated. In addition, an improved Dyer algorithm is proposed and computational experiments show that the improved algorithm is more effective than the original algorithm. © 2005 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 88(8): 38–48, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.20114

Surrogate Programming and Multipliers in Quasi-convex Programming

A result due to Luenberger on the existence of multipliers in a quasi-convex programming problem is extended to the case of constraints given by an arbitrary convex cone under a constraint qualification condition more general than Slater's condition. The existence of solutions is not assumed. We point out links with even convexity in the sense of Fenchel and quasi subdifferentiability in the sense of Greenberg--Pierskalla, and we observe that the couples of primal-dual optimal solutions reduce to saddle-points of a suitable Lagrangian function.

Evaluation of the Improved Dyer Algorithm Characteristics for Determining Surrogate Multipliers of a Surrogate Constraint Method

Evaluation of the Improved Dyer Algorithm Characteristics for Determining Surrogate Multipliers of a Surrogate Constraint Method

A Scatter Search Method for the Bi-Criteria Multi-dimensional {0,1}-Knapsack Problem using Surrogate Relaxation

This paper presents a scatter search (SS) based method for the bi-criteria multi-dimensional knapsack problem. The method is organized according to the usual structure of SS: (1) diversification, (2) improvement, (3) reference set update, (4) subset generation, and (5) solution combination. Surrogate relaxation is used to convert the multi-constraint problem into a single constraint one, which is used in the diversification method and to evaluate the quality of the solutions. The definition of the appropriate surrogate multiplier vector is also discussed. Tests on several sets of large size instances show that the results are of high quality and an accurate description of the entire set of the non-dominated solutions can be obtained within reasonable computational time. Comparisons with other meta-heuristics are also presented. In the tested instances the obtained set of potentially non-dominated solutions dominates the set found with those meta-heuristics.

An Exact Algorithm for the Two-Constraint 0–1 Knapsack Problem

We consider a knapsack problem in which each item has two types of weight and the container has two types of capacity. We discuss the surrogate, Lagrangian, and continuous relaxations, and give an effective method to determine the optimal Lagrangian and surrogate multipliers associated with the continuous relaxation of the problem. These results are used to obtain an exact branch-and-bound algorithm, which also includes heuristic procedures and a reduction technique. The performance of bounds and algorithms is evaluated through extensive computational experiments.

AN IMPROVED SURROGATE CONSTRAINTS METHOD FOR SEPARABLE NONLINEAR INTEGER PROGRAMMING

An improved surrogate constraints method for solving separable nonlinear integer programming problems with multiple constraints is presented. The surrogate constraints method is very effective in solving problems with multiple constraints. The method solves a succession of surrogate constraints problems having a single constraint instead of the original multiple constraint problem. A surrogate problem with an optimal multiplier vector solves the original problem exactly if there is no duality gap. However, the surrogate constraints method often has a duality gap, that is it fails to find an exact solution to the original problem. The modification proposed closes the surrogate duality gap. The modification solves a succession of target problems that enumerates all solutions hitting a particular target. The target problems are produced by using an optimal surrogate multiplier vector. The computational results show that the modification is very effective at closing the surrogate gap of multiple constraint problems.

A simple procedure for solving a continuous quadratic mathematical model

A new efficient procedure for solving a class of continuous linearly constrained quadratic problems is presented. It is based on the use of surrogate multipliers to construct an equivalent knapsack problem that is easier to solve, and a greedy-like procedure to solve it. This procedure is applicable in developing efficient branch and bound algorithms, by computing upper (lower) bounds for a quadratic integer problem.

Smart greedy procedure for solving a multidimensional nonlinear knapsack class of reliability optimization problems

A heuristic procedure called Smart Greedy is proposed for solving a nonlinear knapsack class of reliability optimization problems with multiple constraints (multidimensional nonlinear knapsack problems). At first, by using a surrogate multiplier, the multidimensional nonlinear knapsack problem is translated into an one-dimensional nonlinear knapsack problem, which is called the surrogate problem. Second, modular approach (MA) solves the surrogate problem with the surrogate multiplier given as a centroid of the current polyhedron. Algorithm cut-off polyhedron (COP) provides a cutting plane of the polyhedron, and reduces the polyhedron recursively until the polyhedron becomes empty. Finally, a procedure called Smart Greedy generates an approximate solution of the surrogate problem with the surrogate multiplier finally obtained. The solution obtained is called Smart Greedy solution, which is feasible to the original problem. The computational experiments show hat the present algorithm provides high quality solutions.

Duality-Based Algorithms for Scheduling Unrelated Parallel Machines

We consider the following parallel machine scheduling problem. Each of n independent jobs has to be scheduled on one of m unrelated parallel machines. The processing of job Ji on machine Mi requires an uninterrupted period of positive length pij. The objective is to find an assignment of jobs to machines so as to minimize the maximum job completion time. The objective of this paper is to design practical algorithms for this NP-hard problem. We present an optimization algorithm and an approximation algorithm that are both based on surrogate relaxation and duality. The optimization algorithm solves quite large problems within reasonable time limits. The approximation algorithm is based upon a novel concept for iterative local search, in which the search direction is guided by surrogate multipliers. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A surrogate and Lagrangian approach to constrained network problems

This paper presents the use of surrogate constraints and Lagrange multipliers to generate advanced starting solutions to constrained network problems. The surrogate constraint approach is used to generate a singly constrained network problem which is solved using the algorithm of Glover, Karney, Klingman and Russell [13]. In addition, we test the use of the Lagrangian function to generate advanced starting solutions. In the Lagrangian approach, the subproblems are capacitated network problems which can be solved using very efficient algorithms. The surrogate constraint approach is implemented using the multiplier update procedure of Held, Wolfe and Crowder [16]. The procedure is modified to include a search in a single direction to prevent periodic regression of the solution. We also introduce a reoptimization procedure which allows the solution from thekth subproblem to be used as the starting point for the next surrogate problem for which it is infeasible once the new surrogate constraint is adjoined. The algorithms are tested under a variety of conditions including: large-scale problems, number and structure of the non-network constraints, and the density of the non-network constraint coefficients. The testing clearly demonstrates that both the surrogate constraint and Langrange multipliers generate advanced starting solutions which greatly improve the computational effort required to generate an optimal solution to the constrained network problem. The testing demonstrates that the extra effort required to solve the singly constrained network subproblems of the surrogate constraints approach yields an improved advanced starting point as compared to the Lagrangian approach. It is further demonstrated that both of the relaxation approaches are much more computationally efficient than solving the problem from the beginning with a linear programming algorithm.

A MAXIMUM ENTROPY APPROACH TO CONSTRAINED NON-LINEAR PROGRAMMING

The paper explores the use of the Shannon (informational) entropy measure and Jaynes's maximum entropy formalism in the solution of constrained non-linear programming problems. Through a surrogate constraint approach an entropy based update formula for the surrogate multipliers is derived. A numerical example of the method is presented. Some information-theoretic interpretations of mathematical programming are explored. Finally, through the use or surrogate duals the method is extended into an entropy augmented Lagrangean formulation.

Surrogate duality for vector optimization

Using our theorems (of [12]) on separation of convex sets by linear operators, in the sense of the lexi-cographical order on Rn, we prove some theorems of surrogate duality for vector optimization problems with convex constraints (but no regularity assumption), where the surrogate constraint sets are generalized half-spaces and the surrogate multipliers are linear operators, or isomorphisms, or isometries. In the cae of inequality constraints, we prove that the surrogate multipliers can be taken lexicographically non-negative isometries or non-negative (in the usual order) linear isomorphisms.

Geometric programming with multiplicative slack variables

AbstractA posynomial geometric programming problem formulated so that the number of objective function terms is equal to the number of primal variables will have a zero degree of difficulty when augmented by multiplying each constraint term by a slack variable and including a surrogate constraint composed of the product of the slack variables, each raised to an undetermined negative exponent or surrogate multiplier. It is assumed that the original problem is canonical. The exponents in the constraint on the product of the slack variables must be estimated so that the associated solution to the augmented problem, obtained immediately, also solves the original problem. An iterative search procedure for finding the required exponents, thus solving the original problem, is described. The search procedure has proven quite efficient, often requiring only two or three iterations per degree of difficulty of the original problem. At each iteration the well‐known procedure for solving a geometric programming problem with a zero degree of difficulty is used and so computations are simple. The solution generated at each iteration is optimal for a problem which differs from the original problem only in the values of some of the constraint coefficients, so intermediate solutions provide useful information.

Zero-one integer programs with few constraints —Lower bounding theory

The zero-one integer programming problem and its special case, the multiconstraint knapsack problem frequently appear as subproblems in many combinatorial optimization problems. We present several methods for computing lower bounds on the optimal solution of the zero-one integer programming problem. They include Lagrangean, surrogate and composite relaxations. New heuristic procedures are suggested for determining good surrogate multipliers. Based on theoretical results and extensive computational testing, it is shown that for zero-one integer problems with few constraints surrogate relaxation is a viable alternative to the commonly used Lagrangean and linear programming relaxations. These results are used in a follow up paper to develop an efficient branch and bound algorithm for solving zero-one integer programming problems.

Surrogate Constraints Algorithm for Reliability Optimization Problems with Multiple Constraints

This paper presents a surrogate constraints algorithm for solving nonlinear programming, nonlinear integer programming, and nonlinear mixed integer programming problems. The algorithm contains a new technique for generating a succession of vector values of surrogate multiplier (ie, surrogate problems). By using this technique, a computer can keep a polyhedron, which is a vector space of surrogate multipliers to be considered at a certain time, in its memory. Furthermore it can cut the polyhedron by a given hyperplane, and produce the remaining space as the next polyhedron. Simple examples are included.