Quadratic Binary Programming with Application to Capital-Budgeting Problems

Abstract

The purpose of this paper is to present an algorithm for solving the quadratic binary programming problem. Although a problem with this structure may arise in many situations, it is particularly common in capital budgeting when a decision-maker is confronted with a set of investment proposals from which he must select a portfolio. If returns of proposals are intercorrelated random variables and if the decision-maker uses as his criterion for selection the mean μ and variance σ2 of portfolio returns, his decision requires prior identification of the (μ, σ2) efficient set. The algorithm developed to solve the problem and hence necessary to generate the efficient set is based on the concept of implicit enumeration recently introduced by Egon Balas for solution of the binary linear programming problem.

Local search heuristics for Quadratic Unconstrained Binary Optimization (QUBO)

“…Numerous hard combinatorial optimization problems can be formulated as pseudoBoolean optimization problems, including VLSI design (Barahona et al, 1988;Boros and Hammer, 1991;Boros et al, 1999;Bushnell and Shaik, 1995;Jünger et al, 1994;Krarup andPruzan, 1978 andShaik, 1996), statistical mechanics (Barahona et al, 1988;Simone et al, 1995 andPhillips andRosen, 1994), reliability theory and statistics (Papaioannou, 1977;Ranyard, 1976 andRao, 1971), economics and finance (Hammer and Shliffer, 1971;Hillier, 1969;Laughhunn, 1970;Laughhunn andPeterson, 1971 andMcBride andYormark, 1980), operations research and management science (Gallo et al, 1980;Hammer, 1968;Picard and Ratliff, 1978;Warszawski, 1974 andWeingartner, 1966), manufacturing (Alidaee et al, 1994;Badics, 1996;Badics and Boros, 1998;Crama andMazzola, 1995 andKubiak, 1995), as well as numerous algorithmic problems of discrete mathematics (Fraenkel and Hammer, 1984;Hammer, 1977;Hammer and Rudeanu, 1968;Pardalos and Rodgers, 1992;Picard and Ratliff, 1975;Pardalos and…”

Bundle recommendation in ecommerce

“…One simple application of QKP is the Max-Clique problem [10] where the cross-dependency is represented as unweighted edge in graph. A more practical application of QKP is portfolio selection problem in finance where the cross-dependency is the covariance of two securities [13]. Although QKP is powerful optimization model, it is rarely used for large-scale problem in practice because QKP is NP-hard in general and only small or medium scale problem is tractable by modern mixed integer solver [14,17].…”

Greedy, genetic, and greedy genetic algorithms for the quadratic knapsack problem

“…QKP arises in a variety of applications, including finance [17], VLSI design [7], and location problems [24]. Also, various problems in graphs can be presented as instances of QKP [13] [21].…”