A note on composite concave quadratic programming

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Abstract

In this paper we present a pivotal-based algorithm for the global minimization of composite concave quadratic functions subject to linear constraints. It is shown that certain subclasses of this family yield easy-to-solve line search subproblems. Since the proposed algorithm is equivalent in efficiency to a standard parametric complementary pivoting procedure, the implication is that conventional parametric quadratic programming algorithms can now be used as tools for the solution of much wider class of complex global optimization problems.

Introduction

Over the years various aspects of Quadratic Programming have been the focus of extensive research efforts. Quadratic Programming problems, i.e. the problems of minimizing a quadratic function subject to linear constraints, are very often encountered in various fields of Operations Research 2, 24, 5. Also, advances in Quadratic Programming solution techniques made it possible to utilize them in the solution of general nonlinear programming problems.

Like in convex programming [12], linear fractional programming [4], linear multiobjective programming 8, 23, 25, linear goal programming [7], and composite concave linear programming 22, 15, Dantzig’s pivotal Simplex Method was used effectively for developing algorithms for Quadratic Programming. Pivotal based techniques are either used in a direct manner 2, 24, 5, or in the context of successive approximation [9] and Linear Complementarity 16, 5.

In this paper we present a pivotal based technique for the minimization of a composite concave quadratic objective function subject to linear constraints.

Since generally such a function is neither convex nor concave, optimization problems of this kind are difficult global optimization problems. The development of the method presented in this paper was inspired by the composite concave linear programming method 22, 15 and in fact can be regarded as an extension of it. This should not come as a surprise because both are specific instances of composite concave programming 20, 21, 22 and as such are parametric in nature. Since both linear programming and quadratic programming have powerful parametric analysis tools of the kind needed by composite concave programming algorithms, they are both natural candidates for incorporation in composite programming algorithms as well as other parametric methods for global optimization [13].

The proposed method thus further expands the scope of operation of the good old simplex method and its powerful parametric analysis tools.

Section snippets

Problem formulation

Consider the following composite concave quadratic programming problem:ProblemP:p∗≔minx∈Xf(x)≔q(x)+Φ(l(x))where functions q and l are defined as follows:q(x):=12xTQx+cTx,l(x):=dTx,where c∈n,d∈n are column-vectors, Q∈n×n is a positive-definite matrix; Φ:→ is a real-valued concave function, X≔{x∈n:Ax⩽b,x⩾0} is nonempty and denotes the real line. Let X∗ denote the set of global optimal solutions to Problem P. We shall also consider the case where Φ is applied to the function q, rather than l.

We

Composite concave programming

Adopting a Composite Concave Programming approach 20, 21, 22, we consider the following parametric problem:ProblemP(μ):p∗(μ):=minx∈Xq(x)+μl(x)=minx∈X12xTQx+cTx+μdTx,μ⩽μ⩽μ.Let x∗(μ) denote the unique optimal solution to Problem P(μ) for every fixed value of μ, observing that Q is a positive-definite matrix. This solution is a piecewise affine function of μ. Observe also that μ is a bounded scalar, and the role of bounds will be clarified below. The following is an important useful property of

Parametric analysis

Problem P(μ) is the standard Parametric Quadratic Programming problem described in 2, 24, 16. There are several similar techniques for handling such problems, and there is also commercial software for this purpose (LINDO [19] and OSL IBM [17]). Our analysis is based on the implementation of a parametric complementary pivoting Lemke-type method for Parametric Quadratic Programming 16, 5. For the purposes of this discussion it suffices to say that this method is an adaptation of the very familiar

Special cases

In this section we implement the proposed strategy for some widely used classes of Φ. Note that, for every fixed segment Mi=[μ′,μ″], the objective function Θ(μ) may achieve its minimal value either on one of the end points of the segment, or at the point(s) where Θ′(μ)=0, which from Eq. (13)implies2γμ+δ+ρΦ′(z)|z=ρμ+σ=0.Furthermore, if Φ is twice differentiable, determining the critical values of μ as outlined in the Second-Order Condition involves solving2γ+ρ2Φ″(z)|z=ρμ+σ=0.

From the definition

Illustrative example

Consider the following Generalized Quadratic programming problem:p:=minx∈X1.45x12+0.75x22+7x32−2x1x3−2x2x3+t(x1+x2+x3)p,0⩽p⩽1,where X is the convex set defined by2x1−x2−x3⩾−1,3x2+x3⩽9,3x1+1.5x2+x3⩽12,3x1−3x2+x3⩽3,x1−x2+2x3⩽3.5,2x1+3x2−x3⩾7,x1,x2,x3⩾0.

To comply with the format for Problem P suggested by Eq. (1), let Φ(z)≔tzp. Assign p=1/2, t=2. The resulting parametric concave composite programming problem can be written in the following way:p(μ):=minx∈X1.45x12+0.75x22+7x32−2x1x3−2x2x3+μ(x1+x2+x3

Related problem

The similar problem where the composite function is applied to the function q rather than l, is also amenable to the techniques discussed above. Consider the problemProblemP:π∗:=minx∈Xg(x):=Ψ(q(x))+l(x).The parametric problem in this case is of the following form:ProblemP(μ):π∗(μ):=minx∈Xμq(x)+l(x),μ⩽μ⩽μ.Provided that all the conditions imposed by the Theorem 1 are satisfied and μ⩾0, all the techniques described above for the previous problem are also valid for this problem (subject to division

Acknowledgements

This work was partially supported by the Cooperative Research Centre for Intelligent Decision Systems, Melbourne, Australia. The authors wish to thank the anonymous referee for his/her constructive comments.

References (25)

  • M. Sniedovich

    Algorithmic and computational aspects of composite concave programming

    Int. Trans. Oper. Res.

    (1994)
  • N.S. Bakhvalov, N.P. Zhydkov, G.M. Kobel’kov, Numerical Methods, Nauka, Moscow, 1987 (in...
  • J.C.G. Boot, Quadratic Programming: Algorithms, Anomalies, Applications, North-Holland, Amsterdam,...
  • A. Byrne, M. Sniedovich, L. Churilov, Handling soft constraints via composite concave programming, Journal of the...
  • A. Charnes et al.

    Programming with linear fractional functionals

    Naval Res. Logist. Quart.

    (1962)
  • R.W. Cottle, Jong-Shi Pang, R.E. Stone, The Linear Complementarity Problem, Academic Press, London,...
  • R.W. Cottle, G.B. Dantzig, in: G.B. Dantzig, A.F. Veinott Jr. (Eds.), Complementary pivot theory of mathematical...
  • J.P. Dauer et al.

    An iterative approach to goal programming

    Oper. Res. Quart.

    (1977)
  • T. Gal, Postoptimality Analyses, Parametric Programming, and Related Topics, McGraw-Hill, London,...
  • M. Frank et al.

    An algorithm for quadratic programming

    Naval Res. Logist. Quart.

    (1956)
  • A.M. Geoffrion

    Strictly concave parametric programming, Part I: basic theory

    Management Sci.

    (1966)
  • A.M. Geoffrion

    Strictly concave parametric programming, Part II: additional theory and computational considerations

    Management Sci.

    (1967)

    • Cited by (4)

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