Maximizing for the sum of ratios of two convex functions over a convex set☆
Introduction
The fractional programming is one of the most successful fields today in nonlinear optimization problems. The sum of ratios (SOR) problem is a special class of optimization between fractional programs, and has attracted the interest of practitioners and researchers for at least 30 years. This is because, from a practical point of view, the (SOR) has a number of important applications. Included among these are applications in areas such as transportation planning, government contracting, economics and finance [1], [2], [3], [4]. From a research point of view, the (SOR) poses significant theoretical and computational difficulties. Even in the simplest case where the ratios are all linear, i.e., the numerators and denominators are affine functions, their sum is neither quasiconvex nor quasiconcave though each of them has both properties. This is mainly due to the fact that the (SOR) is a global optimization problem, i.e., it is known to generally possess multiple local optima that are not globally optimal. The sum of ratios problem therefore fall into the domain of global optimization [5], [6]. One can find the details of this development in Refs. [7], [8], [9] and the corresponding bibliographies appearing therein.
Many global optimization algorithms have been proposed for solving the linear sum of ratios fractional programs, i.e., the numerators and denominators are all affine functions and the feasible region is a polyhedron (see [10], [11], [12], [13], for example). Recently, some solution algorithms have been developed for solving globally the nonlinear sum of ratios problem. For instance, Freund and Jarre [14] proposed an interior-point approach for the convex–concave ratios with convex constraints; Yang et al. [15] presented a conical partition algorithm for the sum of DC ratios; Benson [16], [17] gave two branch-and bound algorithms for the concave–convex ratios; Shen et al. [18], [19], [20], [21] developed global optimization algorithms for the nonlinear sum of ratios problem.
In this paper we consider the following sum of ratios problem: where fj(x) and gj(x), , are real-valued convex functions defined on X, X is a nonempty, compact convex set in Rn, and, for each and for all .
The problem (P) is called a nonconcave fractional program, and may arise in practical applications, for instance, the maximally predictable portfolio problem (m=1) [22], and the projective geometry problems including multiview triangulation, camera resectioning and homography estimation [23]. It should be noted that although the literature on nonconvex optimization has rapidly increased in recent years, most research papers either only deal with the theoretical aspects of the problem or are concerned only with finding Kuhn–Tucker points or local solutions rather than global optima. Also, the problem (P) is different from the problems considered in [13], [14], [15], [16], [17], [18], [19], [20], [21]. Specially, the feasible set and each numerator in the objective function in (P) are not the same as the corresponding ones in [21], although both problem (P) and the one considered in [21] are the sum of convex–convex ratios problem. So it is difficult to apply these solution methods in the above Refs. [13], [14], [15], [16], [17], [18], [19], [20], [21] directly to problem (P) since their algorithms are based on the oneself structure of the corresponding problems. As a result, to my knowledge, most of the theoretical and algorithmic work for problem (P) applies only to single-ratio cases, i.e., m=1 (see [24], [25], [26], [27]) or to special cases of (P) (see [10], [11], [12], [13], for example). Additionally, Frenk and Schaible [28] and Schaible [9] have encouraged that more research be done into the solutions of nonconcave fractional programs.
The purpose of this paper is to present a branch and bound algorithm for globally solving the nonlinear sum of ratios problem (P), which does not require involving all the functions to be differentiable and requires that their sub-gradients can be calculated efficiently. In the algorithm, the linear bounding functions related to the objective and constraint functions of (P) are first formed, based on the characteristics of problem (P). Thus a linear relaxation programming is then derived for providing an upper bound of the optimal value to problem (P). The main computational effort of the algorithm only involves in solving a sequence of linear programming subproblems that do not grow in size from iteration to iteration. Additionally, the algorithm uses simplices, rather than more complicated polytopes, as partition elements in the branch and bound search. This keeps the number of constraints in the linear program subproblems to minimum. Finally, the convergence of the algorithm is proved and numerical experiments are given to illustrate the feasibility of the algorithm.
The remainder of this paper is organized as follows. The initial simplex and simplicial partition process, the upper and lower bounding process and the fathoming process used in this approach are defined and studied in Section 2. Section 3 presents the algorithm for solving (P), and shows the convergence property of the algorithm. In Section 4, we give the results of solving some numerical examples with the algorithm.
Section snippets
Key algorithm processes
In this section, in order to present the branch and bound algorithm for solving (P), we first explain the four key processes: successively refined partitioning of the feasible set; estimation of upper and lower bounds for the optimal value of the objective function; and deleting procedure over each subset generated by the partitions.
The partition process consists in a successive simplicial partition of the initial simplex S0 following in an exhaustive subdivision rule, i.e., such that any
Algorithm statement
Based upon the results and algorithmic processes discussed in Section 2, the basic steps of the branch and bound algorithm are summarized in the following. In the algorithm statement, a optimal solution xk for problem (P) refers to a solution such that Branch and bound algorithm.
Step 0: Initialization. Choose such that . Find an n-simplex such that . Solve problem to obtain the optimal value and the optimal solution .
Numerical experiments
To verify the performance of the proposed global optimization algorithm, we give the computational results of the several test examples and the randomly generated problems. The algorithm is coded in Fortran 95, the test problems are implemented on a Pentium (R) 4 CPU 2.00 GHz with 512 MB memory microcomputer. The simplex method is applied to solve the subproblems linear programs in the upper bound process. Numerical results show that the proposed algorithms can globally solve the problem
Concluding remarks
This paper presents a simplicial branch and bound algorithm for globally solving the sum of convex–convex ratios problem. The algorithm computes the upper bounds called for during the branch and bound search by solving linear programming subproblems. These problems are derived by using the well-known concave envelope and convex underestimation method. Computational results for the randomly generated test problems have been reported to show the performance of the proposed algorithm.
We believe
Acknowledgments
The authors are grateful to the responsible editor and the anonymous referees for their valuable comments and suggestions, which have greatly improved the earlier version of this paper.
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Research supported by National Natural Science Foundation of China (11171094) and Fundamental and Frontier Technology Research Projects of Henan Province (132300410285).