A finite algorithm for concave minimization over a polyhedron

We present a new algorithm for solving the problem of minimizing a nonseparable concave function over a polyhedron. The algorithm is of the branch-and-bound type. It finds a globally optimal extreme point solution for this problem in a finite number of steps. One of the major advantages of the algorithm is that the linear programming subproblems solved during the branch-and-bound search each have the same feasible region. We discuss this and other advantages and disadvantages of the algorithm. We also discuss some preliminary computational experience we have had with our computer code for implementing the algorithm. This computational experience involved solving several bilinear programming problems with the code.     

Variations on a cutting plane method for solving concave minimization problems with linear constraints

A cutting plane method for solving concave minimization problems with linear constraints has been advanced by Tui. The principle behind this cutting plane has been applied to integer programming by Balas, Young, Glover, and others under the name of convexity cuts. This paper relates the question of finiteness of Tui's method to the so-called generalized lattice point problem of mathematical programming and gives a sufficient condition for terminating Tui's method. The paper then presents several branch-and-bound algorithms for solving concave minimization problems with linear constraints with the Tui cut as the basis for the algorithm. Finally, some computational experience is reported for the fixed-charge transportation problem.     

An interactive branch-and-bound algorithm for bicriterion nonconvex/mixed integer programming

Although there has been extensive research on interactive multiple objective decision making in the last two decades, there is still a need for specialized interactive algorithms that exploit the relatively simple structure of bicriterion programming problems. This article develops an interactive branch-and-bound algorithm for bicriterion nonconvex programming problems. The algorithm searches among only the set of nondominated solutions since one of them is a most preferred solution that maximizes the overall value function of the decision maker over the set of achievable solutions. The interactive branch-and-bound algorithm requires only pairwise preference comparisons from the decision maker. Based on the decision maker's responses, the algorithm reduces the set of nondominated solutions and terminates with his most preferred nondominated solution. Branching corresponds to dividing the subset of nondominated solutions considered at a node into two subsets. The incumbent solution is updated based on the preference of the decision maker between two nondominated solutions. Fathoming decisions are based on the decision maker's preference between the incumbent solution and the ideal solution of the node in consideration.     

An algorithm and new penalties for concave integer minimization over a polyhedron

We present a branch-and-bound algorithm for globally minimizing a concave function over linear constraints and integer variables. Concave cost functions and integer variables arise in many applications, such as production planning, engineering design, and capacity expansion. To reduce the number of subproblems solved during the branch-and-bound search, we also develop a framework for computing new and existing penalties. Computational testing indicates that penalties based on the Tuy cutting plane provide large decreases in solution time for some problems. A combination of Driebeek-Tomlin and Tuy penalties can provide further decreases in solution time. © 1994 John Wiley & Sons, Inc.     

Surrogate duality in a branch-and-bound procedure

Recent research has led to several surrogate multiplier search procedures for use in a primal branch-and-bound procedure. As single constrained integer programming problems, the surrogate subproblems are also solved via branch-and-bound. This paper develops the inner play between the surrogate subproblem and the primal branch-and-bound trees which can be exploited to produce a number of computational efficiencies. Most important is a restarting procedure which precludes the need to solve numerous surrogate subproblems at each node of a primal branch-and-bound tree. Empirical evidence suggests that this procedure greatly reduces total computation time.     

Convex envelope results and strong formulations for a class of mixed-integer programs

In this article we present a novel technique for deriving the convex envelope of certain nonconvex fixed-charge functions of the type that arise in several related applications that have been considered in the literature. One common attribute of these problems is that they involve choosing levels for the undertaking of several activities. Two or more activities share a common resource, and a fixed charge is incurred when any of these activities is undertaken at a positive level. We consider nonconvex programming formulations for these problems in which the fixed charges are expressed in the form of concave functions. With the use of the developed convex envelope results, we show that the convex envelope relaxations of the nonconvex formulations lead to the linear programming relaxations of the strong IP/MIP formulations of these problems. Moreover, our technique for deriving convex envelopes offers a useful construct that could be exploited in other related contexts as well. © 1996 John Wiley & Sons, Inc.     

An easy solution for a special class of fixed charge problems

The fixed charge problem is a mixed integer mathematical programming problem which has proved difficult to solve in the past. In this paper we look at a special case of that problem and show that this case can be solved by formulating it as a set-covering problem. We then use a branch-and-bound integer programming code to solve test fixed charge problems using the setcovering formulation. Even without a special purpose set-covering algorithm, the results from this solution procedure are dramatically better than those obtained using other solution procedures.     

Concave envelopes of monomial functions over rectangles

The construction of convex and concave envelopes of real‐valued functions has been of interest in mathematical programming for over 3 decades. Much of this interest stems from the fact that convex and concave envelopes can play important roles in algorithms for solving various discrete and continuous global optimization problems. In this article, we use a simplicial subdivision tool to present and validate the formula for the concave envelope of a monomial function over a rectangle. Potential algorithmic applications of this formula are briefly indicated. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

An algorithm for optimal shipments with given frequencies

This article deals with the problem of minimizing the transportation and inventory cost associated with the shipment of several products from a source to a destination, when a finite set of shipping frequencies is available. A mixed-integer programming model—shown to be NP-hard—is formulated for that problem. The computational complexity of some similar models applied to different problems is also investigated. In particular, whereas the capacitated plant location problem with operational cost in product form is NP-hard, the simple plant location problem with the same characteristics can be solved in polynomial time. A branch-and-bound algorithm is finally worked out, and some computational results are presented. © 1996 John Wiley & Sons, Inc.     

A new surrogate dual multiplier search procedure

Efficient computation of tight bounds is of primary concern in any branch-and-bound procedure for solving integer programming problems. Many successful branch-and-bound approaches use the linear programming relaxation for bounding purposes. Significant interest has been reported in Lagrangian and surrogate duals as alternative sources of bounds. The existence of efficient techniques such as subgradient search for solving Lagrangian duals has led to some very successful applications of Lagrangian duality in solving specially structured problems. While surrogate duals have been theoretically shown to provide stronger bounds, the difficulty of surrogate dual-multiplier search has discouraged their employment in solving integer programs. Based on the development of a new relationship between surrogate and Lagrangian duality, we suggest a new strategy for computing surrogate dual values. The proposed approach allows us to directly use established Lagrangian search methods for exploring surrogate dual multipliers. Computational experience with randomly generated capital budgeting problems validates the economic feasibility of the proposed ideas.     

Geometry and resolution of duality gaps

In this study we interpret the exterior penalty function method as a generalized lagrangian metliod which fills duality gaps in nonconvex problems. Geometry and resolution of these gaps from a duality point of view are highlighted.     

A branch-and-bound algorithm for solving fixed charge problems

Numerous procedures have been suggested for solving fixed charge problems. Among these are branch-and-bound methods, cutting plane methods, and vertex ranking methods. In all of these previous approaches, the procedure depends heavily on the continuous costs to terminate the search for the optimal solution. In this paper, we present a new branch-and-bound algorithm that calculates bounds separately on the sum of fixed costs and on the continuous objective value. Computational experience is shown for various standard test problems as well as for randomly generated problems. These test results are compared to previous procedures as well as to a mixed integer code. These comparisons appear promising.     

Capacity improvement,penalties, and the fixed charge transportation problem

Capacity improvement and conditional penalties are two computational aides for fathoming subproblems in a branch‐and‐bound procedure. In this paper, we apply these techniques to the fixed charge transportation problem (FCTP) and show how relaxations of the FCTP subproblems can be posed as concave minimization problems (rather than LP relaxations). Using the concave relaxations, we propose a new conditional penalty and three new types of capacity improvement techniques for the FCTP. Based on computational experiments using a standard set of FCTP test problems, the new capacity improvement and penalty techniques are responsible for a three‐fold reduction in the CPU time for the branch‐and‐bound algorithm and nearly a tenfold reduction in the number of subproblems that need to be evaluated in the branch‐and‐bound enumeration tree. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 341–355, 1999     

A specially structured nonlinear integer resource allocation problem

We present an algorithm for solving a specially structured nonlinear integer resource allocation problem. This problem was motivated by a capacity planning study done at a large Health Maintenance Organization in Texas. Specifically, we focus on a class of nonlinear resource allocation problems that involve the minimization of a convex function over one general convex constraint, a set of block diagonal convex constraints, and bounds on the integer variables. The continuous variable problem is also considered. The continuous problem is solved by taking advantage of the structure of the Karush‐Kuhn‐Tucker (KKT) conditions. This method for solving the continuous problem is then incorporated in a branch and bound algorithm to solve the integer problem. Various reoptimization results, multiplier bounding results, and heuristics are used to improve the efficiency of the algorithms. We show how the algorithms can be extended to obtain a globally optimal solution to the nonconvex version of the problem. We further show that the methods can be applied to problems in production planning and financial optimization. Extensive computational testing of the algorithms is reported for a variety of applications on continuous problems with up to 1,000,000 variables and integer problems with up to 1000 variables. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 770–792, 2003.     

A descent approach to solving the complementary programming problem

In recent years, much attention has focused on mathematical programming problems with equilibrium constraints. In this article we consider the case where the constraints are complementarity constraints. Problems of this type arise, for instance, in the design of traffic networks. We develop here a descent algorithm for this problem that will converge to a local optimum in a finite number of iterations. The method involves solving a sequence of subproblems that are linear programs. Computational tests comparing our algorithm with the branch-and-bound algorithm in [7] bear out the efficacy of our method. When solving large problems, there is a definite advantage to coupling both methods. A local optimum incumbent provided by our algorithm can significantly reduce the computational effort required by the branch-and-bound algorithm.     

Some branch-and-bound procedures for fixed-cost transportation problems

In this article we present three properties that will improve the performance of branch-and-bound algorithms for fixed-cost transportation problems. By applying Lagrangian relaxation we show that one can develop stronger up and down penalties than those traditionally used and also develop a strengthened penalty for nonbasic variables. We also show that it is possible to “look ahead” of a particular node and determine the solution at the next node without actually calculating it. We present computational evidence by comparing our developments with existing procedures.     

Transportation type problems with quantity discounts

It is known to be real that the per unit transportation cost from a specific supply source to a given demand sink is dependent on the quantity shipped, so that there exist finite intervals for quantities where price breaks are offered to customers. Thus, such a quantity discount results in a nonconvex, piecewise linear functional. In this paper, an algorithm is provided to solve this problem. This algorithm, with minor modifications, is shown to encompass the “incremental” quantity discount and the “fixed charge” transportation problems as well. It is based upon a branch-and-bound solution procedure. The branches lead to ordinary transportation problems, the results of which are obtained by utilizing the “cost operator” for one branch and “rim operator” for another branch. Suitable illustrations and extensions are also provided.     

Models for multi-item continuous review inventory policies subject to constraints

Models are formulated for determining continuous review (Q, r) policies for a multiitem inventory subject to constraints. The objective function is the minimization of total time-weighted shortages. The constraints apply to inventory investment and reorder workload. The formulations are thus independent of the normal ordering, holding, and shortage costs. Two models are presented, each representing a convex programming problem. Lagrangian techniques are employed with the first, simplified model in which only the reorder points are optimized. In the second model both the reorder points and the reorder quantities are optimized utilizing penalty function methods. An example problem is solved for each model. The final section deals with the implementation of these models in very large inventory systems.     

An interactive paired comparison method for bicriterion integer programming

This article proposes an interactive paired comparison region elimination method for bicriterion integer mathematical programming problems. The new method isolates the best compromise solution by successively evaluating a pair of associated supported non-dominated solutions. The efficiency of the method is tested by solving randomly generated problems based on varying shapes of efficient frontiers. When compared with the existing branch-and-bound method, the method was effective in reducing the burden on the decision maker. © 1994 John Wiley & Sons, Inc.     

Maximizing the sum of certain quasiconcave functions using generalized benders decomposition

In this paper we consider the problem of maximizing the sum of certain quasi-concave functions over a convex set. The functions considered belong to the classes of functions which are known as nonlinear fractional and binonlinear functions. Each individual function is quasi-concave but the sum is not. We show that this nonconvex programming problem can be solved using Generalized Benders Decomposition as developed by Geoffrion.