Nonlinear integer programming for optimal allocation in stratified sampling

A stratified random sampling plan is one in which the elements of the population are first divided into nonoverlapping groups, and then a simple random sample is selected from each group. In this paper, we focus on determining the optimal sample size of each group. We show that various versions of this problem can be transformed into a particular nonlinear program with a convex objective function, a single linear constraint, and bounded variables. Two branch and bound algorithms are presented for solving the problem. The first algorithm solves the transformed subproblems in the branch and bound tree using a variable pegging procedure. The second algorithm solves the subproblems by performing a search to identify the optimal Lagrange multiplier of the single constraint. We also present linearization and dynamic programming methods that can be used for solving the stratified sampling problem. Computational testing indicates that the pegging branch and bound algorithm is fastest for some classes of problems, and the linearization method is fastest for other classes of problems.     

A production-transportation problem with stochastic demand and concave production costs

Well known extensions of the classical transportation problem are obtained by including fixed costs for the production of goods at the supply points (facility location) and/or by introducing stochastic demand, modeled by convex nonlinear costs, at the demand points (the stochastic transportation problem, [STP]). However, the simultaneous use of concave and convex costs is not very well treated in the literature. Economies of scale often yield concave cost functions other than fixed charges, so in this paper we consider a problem with general concave costs at the supply points, as well as convex costs at the demand points. The objective function can then be represented as the difference of two convex functions, and is therefore called a d.c. function. We propose a solution method which reduces the problem to a d.c. optimization problem in a much smaller space, then solves the latter by a branch and bound procedure in which bounding is based on solving subproblems of the form of [STP]. We prove convergence of the method and report computational tests that indicate that quite large problems can be solved efficiently. Problems up to the size of 100 supply points and 500 demand points are solved. Received October 11, 1993 / Revised version received July 31, 1995 Published online November 24, 1998     

Minimum-support solutions of polyhedral concave programs *

Motivated by the successful application of mathematical programming techniques to difficult machine learning problems, we seek solutions of concave minimization problems over polyhedral sets with minimum number of nonzero components. We that if

such problems have a solution, they have a vertex solution with a minimal number of zeros. This includes linear programs and general linear complementarity problems. A smooth concave exponential approximation to a step function solves the minimumsupport problem exactly for a finite value of the smoothing parameter. A fast finite linear-programming-based iterative method terminates at a stationary point, which for many important real world problems provides very useful answers. Utilizing the

complementarity property of linear programs and linear complementarity problems, an upper bound on the number of nonzeros can be obtained by solving a single convex minimization problem on a polyhedral set     

Stochastic Multiobjective Optimization: Sample Average Approximation and Applications

We investigate one stage stochastic multiobjective optimization problems where the objectives are the expected values of random functions. Assuming that the closed form of the expected values is difficult to obtain, we apply the well known Sample Average Approximation (SAA) method to solve it. We propose a smoothing infinity norm scalarization approach to solve the SAA problem and analyse the convergence of efficient solution of the SAA problem to the original problem as sample sizes increase. Under some moderate conditions, we show that, with probability approaching one exponentially fast with the increase of sample size, an ϵ-optimal solution to the SAA problem becomes an ϵ-optimal solution to its true counterpart. Moreover, under second order growth conditions, we show that an efficient point of the smoothed problem approximates an efficient solution of the true problem at a linear rate. Finally, we describe some numerical experiments on some stochastic multiobjective optimization problems and report preliminary results.     

A global supply chain model with transfer pricing and transportation cost allocation

We present a model for the optimization of a global supply that maximizes the after tax profits of a multinational corporation and that includes transfer prices and the allocation of transportation costs as explicit decision variables. The resulting mathematical formulation is a non-convex optimization problem with a linear objective function, a set of linear constraints, and a set of bilinear constraints. We develop a heuristic solution algorithm that applies successive linear programming based on the reformulation and the relaxation of the original problem. Our computational experiments investigate the impact of using different starting points. The algorithm produces feasible solutions with very small gaps between the solutions and their upper bound (UB).     

Multiple reduced gradient method for multiobjective optimization problems

Multiobjective optimization has a large number of real-life applications. Under this motivation, in this paper, we present a new method for solving multiobjective optimization problems with both linear constraints and bound constraints on the variables. This method extends, to the multiobjective setting, the classical reduced gradient method for scalar-valued optimization. The proposed algorithm generates a feasible descent direction by solving an appropriate quadratic subproblem, without the use of any scalarization approaches. We prove that the sequence generated by the algorithm converges to Pareto-critical points of the problem. We also present some numerical results to show the efficiency of the proposed method.     

Inverse integer optimization with an imperfect observation

This study develops and evaluates methods for inverse integer optimization problems with an imperfect observation where the unknown parameters are the cost coefficients. We propose a cutting plane algorithm for this problem and compare it to a heuristic which solves the inverse of the linear relaxation of the forward problem. We then propose a hybrid approach that initializes the cutting plane algorithm from the solution of the heuristic.     

On the convergence of a Jacobi-type algorithm for singly linearly-constrained problems subject to simple bounds

In this work we define a block decomposition Jacobi-type method for nonlinear optimization problems with one linear constraint and bound constraints on the variables. We prove convergence of the method to stationary points of the problem under quite general assumptions.     

Assessing solution quality in stochastic programs

Determining whether a solution is of high quality (optimal or near optimal) is fundamental in optimization theory and algorithms. In this paper, we develop Monte Carlo sampling-based procedures for assessing solution quality in stochastic programs. Quality is defined via the optimality gap and our procedures' output is a confidence interval on this gap. We review a multiple-replications procedure that requires solution of, say, 30 optimization problems and then, we present a result that justifies a computationally simplified single-replication procedure that only requires solving one optimization problem. Even though the single replication procedure is computationally significantly less demanding, the resulting confidence interval might have low coverage probability for small sample sizes for some problems. We provide variants of this procedure that require two replications instead of one and that perform better empirically. We present computational results for a newsvendor problem and for two-stage stochastic linear programs from the literature. We also discuss when the procedures perform well and when they fail, and we propose using ɛ-optimal solutions to strengthen the performance of our procedures.     

A fast heuristic method for minimizing traffic congestion on reconfigurable ring topologies

We describe the development of fast heuristics and methodologies for congestion minimization problems in directional wireless networks, and we compare their performance with optimal solutions. The focus is on the network layer topology control problem (NLTCP) defined by selecting an optimal ring topology as well as the flows on it. Solutions to NLTCP need to be computed in near realtime due to changing weather and other transient conditions and which generally preclude traditional optimization strategies. Using a mixed-integer linear programming formulation, we present both new constraints for this problem and fast heuristics to solve it. The new constraints are used to increase the lower bound from the linear programming relaxation and hence speed up the solution of the optimization problem by branch and bound. The upper and lower bounds for the optimal objective function to the mixed integer problem then serve to evaluate new node-swapping heuristics which we also present. Through a series of tests on different sized networks with different traffic demands, we show that our new heuristics achieve within about 0.5% of the optimal value within seconds.     

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.     

A new approach for solving linear bilevel problems using genetic algorithms

Bilevel programming involves two optimization problems where the constraint region of the first level problem is implicitly determined by another optimization problem. This paper develops a genetic algorithm for the linear bilevel problem in which both objective functions are linear and the common constraint region is a polyhedron. Taking into account the existence of an extreme point of the polyhedron which solves the problem, the algorithm aims to combine classical extreme point enumeration techniques with genetic search methods by associating chromosomes with extreme points of the polyhedron. The numerical results show the efficiency of the proposed algorithm. In addition, this genetic algorithm can also be used for solving quasiconcave bilevel problems provided that the second level objective function is linear.     

Upper Bounds on Height of Terms in a Solution of a Semiunification Problem

In this paper, we investigate a decidable case of the term semiunification problem. In our case, the arity of any functional symbol in terms equals one (with a possible exception for outer symbols of terms, for which no restrictions are imposed). We propose an algorithm constructing the most general solution of the semiunification problem. In addition, we prove an upper bound on the height of this solution; this upper bound is linear with respect to heights of terms in the initial problem. Our method reduces the problem for terms to a special system of equations in free semigroups and solves the latter system. Bibliography: 10 titles.     

Primal and dual algorithms for optimization over the efficient set

ABSTRACT

Optimization over the efficient set of a multi-objective optimization problem is a mathematical model for the problem of selecting a most preferred solution that arises in multiple criteria decision-making to account for trade-offs between objectives within the set of efficient solutions. In this paper, we consider a particular case of this problem, namely that of optimizing a linear function over the image of the efficient set in objective space of a convex multi-objective optimization problem. We present both primal and dual algorithms for this task. The algorithms are based on recent algorithms for solving convex multi-objective optimization problems in objective space with suitable modifications to exploit specific properties of the problem of optimization over the efficient set. We first present the algorithms for the case that the underlying problem is a multi-objective linear programme. We then extend them to be able to solve problems with an underlying convex multi-objective optimization problem. We compare the new algorithms with several state of the art algorithms from the literature on a set of randomly generated instances to demonstrate that they are considerably faster than the competitors.     

Variable-sized uncertainty and inverse problems in robust optimization

In robust optimization, the general aim is to find a solution that performs well over a set of possible parameter outcomes, the so-called uncertainty set. In this paper, we assume that the uncertainty size is not fixed, and instead aim at finding a set of robust solutions that covers all possible uncertainty set outcomes. We refer to these problems as robust optimization with variable-sized uncertainty. We discuss how to construct smallest possible sets of min–max robust solutions and give bounds on their size.A special case of this perspective is to analyze for which uncertainty sets a nominal solution ceases to be a robust solution, which amounts to an inverse robust optimization problem. We consider this problem with a min–max regret objective and present mixed-integer linear programming formulations that can be applied to construct suitable uncertainty sets.Results on both variable-sized uncertainty and inverse problems are further supported with experimental data.     

Preprocessing rules for integer programming solutions to the generalised assignment problem

Several preprocessing rules to reduce integer programming problem size are proposed and examined for the generalised assignment problem. The rules make use of the linear programming relaxation and ranking on the basis of a combined resource/cost metric. Computational experiments with commercial branch and bound solvers have been performed using publicly available problem data from the OR library. Results are promising as the preprocessed problem solves to within 1% of the full problem with significantly less CPU time for most of the test problems examined.     

Split common fixed point and common null point problem

In this paper, we present a new algorithm for solving the split common null point and common fixed point problem, to find a point that belongs to the common element of common zero points of an infinite family of maximal monotone operators and common fixed points of an infinite family of demicontractive mappings such that its image under a linear transformation belongs to the common zero points of another infinite family of maximal monotone operators and its image under another linear transformation belongs to the common fixed point of another infinite family of demicontractive mappings in the image space. We establish strong convergence for the algorithm to find a unique solution of the variational inequality, which is the optimality condition for the minimization problem. As special cases, we shall use our results to study the split equilibrium problems and the split optimization problems.     

Relaxation-based algorithms for minimax optimization problems with resource allocation applications

We consider minimax optimization problems where each term in the objective function is a continuous, strictly decreasing function of a single variable and the constraints are linear. We develop relaxation-based algorithms to solve such problems. At each iteration, a relaxed minimax problem is solved, providing either an optimal solution or a better lower bound. We develop a general methodology for such relaxation schemes for the minimax optimization problem. The feasibility tests and formulation of subsequent relaxed problems can be done by using Phase I of the Simplex method and the Farkas multipliers provided by the final Simplex tableau when the corresponding problem is infeasible. Such relaxation-based algorithms are particularly attractive when the minimax optimization problem exhibits additional structure. We explore special structures for which the relaxed problem is formulated as a minimax problem with knapsack type constraints; efficient algorithms exist to solve such problems. The relaxation schemes are also adapted to solve certain resource allocation problems with substitutable resources. There, instead of Phase I of the Simplex method, a max-flow algorithm is used to test feasibility and formulate new relaxed problems.Corresponding author.Work was partially done while visiting AT&T Bell Laboratories.     

A dual algorithm for the one-machine scheduling problem

A branch and bound algorithm is presented for the problem of schedulingn jobs on a single machine to minimize tardiness. The algorithm uses a dual problem to obtain a good feasible solution and an extremely sharp lower bound on the optimal objective value. To derive the dual problem we regard the single machine as imposing a constraint for each time period. A dual variable is associated with each of these constraints and used to form a Lagrangian problem in which the dualized constraints appear in the objective function. A lower bound is obtained by solving the Lagrangian problem with fixed multiplier values. The major theoretical result of the paper is an algorithm which solves the Lagrangian problem in a number of steps proportional to the product ofn ² and the average job processing time. The search for multiplier values which maximize the lower bound leads to the formulation and optimization of the dual problem. The bounds obtained are so sharp that very little enumeration or computer time is required to solve even large problems. Computational experience with 20-, 30-, and 50-job problems is presented.     

A new discrete filled function method for solving large scale max-cut problems

The global optimization method based on discrete filled function is a new method that solves large scale max-cut problems. We first define a new discrete filled function based on the structure of the max-cut problem and analyze its properties. Unlike the continuous filled function methods, by the characteristic of the max-cut problem, the parameters in the proposed filled function does not need to be adjusted. By combining a procedure that randomly generates initial points for minimization of the proposed filled function, the proposed algorithm can greatly reduce the computational time and be applied to large scale max-cut problems. Numerical results and comparisons with several heuristic methods indicate that the proposed algorithm is efficient and stable to obtain high quality solution of large scale max-cut problems.