Continuous Optimization
Strong duality for robust minimax fractional programming problems

https://doi.org/10.1016/j.ejor.2013.02.015Get rights and content

Abstract

We develop a duality theory for minimax fractional programming problems in the face of data uncertainty both in the objective and constraints. Following the framework of robust optimization, we establish strong duality between the robust counterpart of an uncertain minimax convex–concave fractional program, termed as robust minimax fractional program, and the optimistic counterpart of its uncertain conventional dual program, called optimistic dual. In the case of a robust minimax linear fractional program with scenario uncertainty in the numerator of the objective function, we show that the optimistic dual is a simple linear program when the constraint uncertainty is expressed as bounded intervals. We also show that the dual can be reformulated as a second-order cone programming problem when the constraint uncertainty is given by ellipsoids. In these cases, the optimistic dual problems are computationally tractable and their solutions can be validated in polynomial time. We further show that, for robust minimax linear fractional programs with interval uncertainty, the conventional dual of its robust counterpart and the optimistic dual are equivalent.

Highlights

► New duality theory for minimax fractional programs in the face of data uncertainty. ► Robust optimization approach to duality theory for inexact fractional programs. ► Easily solvable duals for minimax linear fractional programs under uncertainty.

Introduction

A minimax fractional programming problem in the face of data uncertainty both in the objective and constraints can be captured by the parameterized minimax fractional programming model problem(UP)minxRnmax1ipfi(x,ui)gi(x,vi)s.t.hk(x,wk)0,k=1,2,,m,where ui, vi and wk are uncertain parameters (or data) which belong to the convex and compact uncertainty sets Ui, Vi and Wk, for each i=1,2,,p,fi,gi:Rn×RhR and, for each k = 1, 2,  , m, hk:Rn×RhR. This model covers the robust fractional programming model program, studied recently in [16], where p = 1 and fi and gi are free of data uncertainty. On the other hand, minimax fractional programming problems have been extensively studied [1], [3], [8], [9], [10], [11], [25], [26], [27], [29], [30] without taking into account data uncertainty despite the fact majority of optimization problems are often affected by data uncertainty due to modelling or estimation errors [4], [5], [13].

Robust optimization has now emerged as a computationally powerful approach to dealing with data uncertainty in optimization. An excellent recent survey of robust optimization approaches is given in [6], [5]. Following this approach, the robust counterpart of the uncertain problem (UP), which is termed as the robust primal program, is given by(RP)minxRnmax1ipmax(ui,vi)Ui×Vifi(x,ui)gi(x,vi)s.t.hk(x,wi)0,wkWk,k=1,2,,m,where the constraints are enforced for all possible uncertain parameters. The robust counterpart aims at finding a worst-case solution that is immunized against the data uncertainty. However, (RP) is, in general, often a hard problem to solve computationally.

Similarly, for each fixed parameter ui, vi and wk, (UP) has its conventional dual program (see [8], [11]), given bymaxλR+m,yS+p,(x,α)Rn×R+αs.t.i=1pyi[xfi(x,ui)-αxgi(x,vi)]+k=1mλkxhk(x,wk)=0,i=1pyi[fi(x,ui)-αgi(x,vi)]+k=1mλkhk(x,wk)0,where S+p(x1,,xp):xi0,i=1pxi=1 and ∇x denotes the derivative with respect to the first variable. By maximizing over all possible ui  Ui, vi  Vi and wk  Wk, we arrive at the optimistic counterpart [2] of this uncertain dual, called optimistic dual program, which is given by(ODP)maxλR+m,yS+p,(x,α)Rn×R+,(ui,vi,wk)Ui×Vi×Wkαs.t.i=1pyi[xfi(x,ui)-αxgi(x,vi)]+k=1mλkxhk(x,wk)=0,i=1pyi[fi(x,ui)-αgi(x,vi)]+k=1mλkhk(x,wk)0.

Strong duality holds for the robust minimax fractional programming problems whenever the optimal value of the robust primal program (RP) equals the optimal value of the optimistic dual program (ODP) and the maximum of the optimistic dual is attained. Recently, duality results have been established for various classes of robust convex programming problems [2], [15], [21], [22], including robust semi-infinite linear programming problems [12] and semi-definite linear programming problems [17]. Strong duality results have also been given for robust generalized convex programming problems [20] and convex–concave robust fractional programming problems [16].

The purpose of this paper is to establish strong duality between (RP) and (ODP) under suitable convexity and concavity conditions and to provide classes of robust minimax fractional programs for which (ODP) is easily solvable. In particular, we show that the optimistic dual of a robust minimax linear fractional program with scenario uncertainty in the numerator of the objective function is a simple linear program when the constraint uncertainty is expressed as bounded intervals. We also show that the optimistic dual can be reformulated as a second-order cone programming problem when the constraint uncertainty is given in terms of ellipsoids. In these cases, the optimistic dual problems are computationally tractable and their solutions can be validated in polynomial time. We further show that, for robust minimax linear fractional programs, the conventional dual of the robust counterpart and the optimistic dual are equivalent.

The outline of the paper is as follows. Section 2 presents weak and strong duality results for convex–concave minimax fractional programming problems. Section 3 provides computationally tractable classes of optimistic dual problems where uncertainty sets are given in terms of bounded intervals or ellipsoids as commonly used in robust optimization. Section 4 establishes that, for robust minimax linear fractional programs, the conventional dual of the robust counterpart and the optimistic dual are equivalent. Section 5 concludes with a discussion on further research on minimax fractional programming under data uncertainty.

Section snippets

Strong duality

In this section, we present weak and strong duality results for robust minimax fractional programming problems under a strict feasibility condition. Throughout this paper, we assume that, for each i = 1, 2,  , p, Ui, Vi and Wk are convex and compact uncertainty sets. We also assume that a feasible solution of the robust counterpart problem (RP) exists. That is, the feasible set F of (RP), defined byF{xRn|hk(x,wk)0,wkWk,i=1,2,,m}is non-empty.

Theorem 2.1 Weak duality for robust minimax fractional programs

For (RP) and (ODP), for each i = 1, 2,  , p, let fi(·, ui

Minimax linear fractional problems: tractable optimistic duals

In this section, we provide two classes of uncertain minimax fractional linear programming problems where the corresponding optimistic dual problems can be reformulated as simple linear programming problems or second order programming problems, and so, are computationally tractable and their solutions can be validated in polynomial time.

Optimistic dual vs dual of robust counterpart

In this section, we examine the relationship between the optimistic dual and a conventional dual problem of the robust counterpart in the case of minimax linear fractional programming problems with interval uncertainty sets Ui, Vi and Wk. In this case, we show that these two types of dual problems are indeed equivalent.

Consider the following linear minimax fractional programming problem under interval uncertainty:(LP)minxRnmax1ipaiTx+biciTx+dis.t.wkTxγk,k=1,2,,m,where the datum (ai,ci,wk)R

Conclusion and further research

In this paper, using robust optimization approach, we developed a duality theory for minimax fractional programming problems in the face of data uncertainty both in the objective and constraints. In particular, we established strong duality between the robust counterpart of an uncertain minimax convex–concave fractional program, and the optimistic counterpart of its uncertain conventional dual program. We also provided two classes of uncertain minimax linear fractional programming problems

Acknowledgments

The authors are grateful to the referees for their constructive comments and helpful suggestions which have contributed to the final preparation of the paper. Research was partially supported by a Grant from the Australian Research Council.

References (31)

  • A. Ben-Tal et al.

    Robust optimization—methodology and applications

    Math. Program., Ser. B

    (2002)
  • A. Ben-Tal et al.

    Robust Optimization

    Princeton Series in Applied Mathematics

    (2009)
  • D. Bertsimas et al.

    Theory and applications of robust optimization

    SIAM Review

    (2011)
  • F.H. Clarke

    Optimization and Nonsmooth Analysis

    (1983)
  • B.D. Craven

    Fractional Programming

    (1988)
  • Cited by (21)

    • On Mathematical Programs with Equilibrium Constraints Under Data Uncertainty

      2023, Springer Proceedings in Mathematics and Statistics
    View all citing articles on Scopus
    View full text