Linear semi-infinite programming theory: An updated survey

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Abstract

This paper presents an state-of-the-art survey on linear semi-infinite programming theory and its extensions (in particular, convex semi-infinite programming). This review updates a previous survey [Semi-Infinite Programming, Non-convex Optim. Appl. 25, 1998] of the same authors on the same topic which was published in 1998.

Introduction

Every LP problem can be formulated in standard format asSupt=1mbtλts.t.t=1matλt=c,λt⩾0,t=1,2,…,m,where the unknowns are λt, t=1,2,…,m, and the data are c∈Rn, atRn, and btR, t=1,2,…,m. Its corresponding dual problem can be written asInfcxs.t.atx⩾bt,t=1,2,…,m,with unknowns x=(x1,…,xn).

A natural extension of this dual pair can be obtained by replacing the index set {1,2,…,m} with an arbitrary set T (possibly without any topology), and allowing only a finite number of the associated variables, λt, tT, to be non-zero. In other words, the space of the variables of the first problem, Rm, is replaced with the linear space of all the functions λ:T→R such that λt=0 for all tT except maybe for a finite number of indices, the supporting set of λ, that we denote by suppλ={t∈T∣λt≠0}. This linear space is denoted by R(T) (the so-called space of generalized finite sequences), and its positive cone by R+(T). Thus, we associate with the given data, which form the triple (P)=(c,a.,b.), with c∈Rn, a:T→Rn, and b:T→R, the pair of problems(D)Supt∈Tλtbts.t.t∈Tλtat=c,λ∈R+(T),and(P)Infcxs.t.atx⩾bt,forallt∈T.The constraint system of (P), σ={atxbt,tT} (σ can be seen as an external representation of the feasible set), is called a linear semi-infinite system (LSIS in brief), (P) is a primal linear semi-infinite programming (LSIP) problem and (D) is its corresponding (Haar's) dual. If T is an infinite set, then (P) has finitely many variables and infinitely many constraints, whereas the opposite occurs with (D), so that both problems can be properly called semi-infinite. If T is compact and the functions at and bt are continuous on T, then σ and (P) are said to be continuous. We shall denote by F and F* (by Λ and Λ*) the feasible and the optimal set of (P) (of (D), respectively), whereas v (vD) represents the optimal value of (P) (of (D), respectively). An inequality axb is a consequence of σ if it is satisfied for all xF. When different problems are considered, they and their associated elements will be distinguished by means of sub(super)scripts. So, associated with the problem (P1):=(c1,a1,b1), we have F1, Λ1, F1*, Λ1*, etc.

The last two surveys on LSIP methods and applications were published in 1998 (Section 3 in [77] and Part I in [34], respectively). Since then the simplex method for (D) has been revisited in [8], whereas new methods for the primal problem (P) have been proposed, namely: feasible directions methods [59], [60], cutting plane methods [12], [91], a logarithmic barrier decomposition method with its corresponding complexity analysis [69], a perturbation method [88], and a method providing an element of F* as the limit point of a sequence of optimal solutions of unconstrained convex programming problems [65]. Between the most recent LSIP applications let us mention those dealing with design problems [2], [74], [79], optimization problems of different kinds [3], [11], [55], [68], fuzzy linear programming and systems [27], [42], [61], optimal control [25], approximation problems [46], [72], transportation problems [87], numerical analysis [38], [39], general economic equilibrium [45], games [80], [84], geometry [35], [49], and probability and statistics [13], [22].

Replacing the linear functions in (P) by convex functions we obtain the primal convex semi-infinite programming (CSIP) problem. Numerical methods for CSIP problems with linear constraints have been proposed in [66], [90] (in both cases for quadratic objective functions), and [82] (for a class of problems arising in statistical inference). Concerning CSIP with non-linear constraints, a cutting plane method is given in [42] (on fuzzy convex systems), and two logarithmic barrier methods are developed in [50], [83] (paper which deals with control problems arising in finance). For solving ill-posed CSIP problems, (prox)-regularized path-following and logarithmic barrier methods are respectively proposed in [51], [52]. As an extension of the procedure in the last reference, in [1] a regularized logarithmic barrier method is presented, allowing for an unbounded solution set and for the presence of non-differentiable convex functions in the objective and in the constraints system.

As exposed in our previous survey [33], LSISs theory provides the foundations for the LSIP theory, including the theoretical basis of certain numerical approaches. Thus, Section 2 reviews some recent advances in LSISs theory concerning the geometrical properties of F, and the optimality and duality properties of (P). Section 3 deals with different notions of Hadamard well-posedness in LSIP. These notions are related to the stability of the nominal problem, as well as to the convergence properties of particular sequences of approximating problems. In Section 4 the stability of the LSISs is analyzed in a very broad parametric setting, created through the notion of parametrization mapping. Also in this section some particular approaches to the stability of the feasible set are surveyed. They are mainly based on the behavior of the so-called carrier index set, and on the possibility of approximating the nominal LSIS by finite subsystems. Some work from the “extended” field of stability for the non-linear SIP must be mentioned here. For instance [48], where the feasible set is compact and the parameters are the defining functions; and [76], where the topological stability of the feasible set is characterized, despite its unboundedness. Section 5 is devoted to the CSIP, and the main topics considered there are the constraint qualifications and their implications in stability.

Now, let us introduce some notation. We consider the Euclidean space Rn, whose zero-vector, open unit ball and unit sphere will be denoted by 0n, Bn and Sn, respectively. The canonical basis of Rn shall be represented by e1,…,en. For any set X≠∅, let us denote by spanX, affX, coneX, X0 and X the linear span of X, the affine manifold spanned by X, the convex conical hull of X, the positive polar of X, and the subspace of the vectors that are orthogonal to all the elements of X, respectively. On the topological side, we denote by clX, bdX, rbdX and rintX the closure, the boundary, the relative boundary and the relative interior of X (relative means with respect to the topology of affX). We denote by dimX the dimension of X (i.e., the dimension of affX), by rankX the dimension of spanX, and by O+X the cone of recession directions of X. A vector y∈Rn is a feasible direction at xX if there exists ε>0 such that x+εyX. The cone of feasible directions at x will be denoted by D(X;x). The Euclidean distance from x to X(≠∅) will be denoted by d(x,X).

Section snippets

Geometry, optimality and duality

The following two classes of consistent LSISs played important roles in Sections 3 (geometry), 4 (optimality) and 5 (duality and discretization) of our previous survey [33]: σ is Farkas–Minkowski (FM) when every linear consequent relation of σ is also the consequence of a finite subsystem of σ, and it is said to be locally polyhedral (LOP) if A(x)0=D(F;x) for all xF, where A(x):=cone{at|atx=bt,t∈T} is the so-called active cone at x. Any consistent finite system is both FM and LOP. A new

Solving strategies and well-posedness

Roughly speaking the well-posedness of a problem will be connected with the possibility of properly solving it by means of proximal problems. The concept of solving strategy allows us to offer an unified treatment of different notions of Hadamard well-posedness for the primal LSIP problem (P), when its solution is related to the solutions of the problems in a approximating sequence {(Pr)}. In [19] two strategies are considered which either approximately solve or exactly solve the approximating

Stability of linear inequality systems in a parametric setting

The stability theory for LSIS presented in [18], [34] is connected with the idea of that systems have coefficients whose values are not exactly known, or that they have been rounded off in the computing process. So, we are considering a different system σ1={(at1)x⩾bt1,t∈T} instead of the nominal one, σ={atx⩾bt,t∈T}. In order to measure the size of the perturbations, we restrict to the present parameter space (identified with (Rn×R)T) the extended distance d introduced in the previous section,

Convex semi-infinite programming

In the optimality theory for the ordinary convex optimization problem(CP)Inff(x)s.t.gt(x)⩽0,t∈T={1,2,…,m},where f and gt, tT, are real convex finite-valued functions on Rn, the basic constraint qualification (BCQ, in short) plays an important role. We say that the constraints system σ={gt(x)⩽0,t∈T} satisfies the BCQ ifNF(x)=A(x),forallx∈bdF,where NF(x) is the normal cone to the feasible set F at the point x; i.e.,NF(x):={y∈Rn|y(z−x)⩽0,forallz∈F}≡(x−F)0≡−D(F;x)0,and A(x) extends to the convex

Acknowledgements

This work was supported by the DGES of Spain, Grant PB98-0975. The authors are very grateful to their colleagues M.D. Cánovas, M.D. Fajardo, J. Parra, and to the anonymous referees for their invaluable comments and suggestions.

References (96)

  • I.K. Altinel et al.

    A dynamic model for component testing

    Naval Research Logistics

    (1997)
  • J. Amaya et al.

    Strong duality for inexact linear programming problems

    Optimization

    (2001)
  • E.J. Anderson et al.

    Simplex-like trajectories on quasi-polyhedral convex sets

    Mathematics of Operations Research

    (2001)
  • E.J. Anderson et al.

    An extension of the simplex algorithm for semi-infinite linear programming

    Mathematical Programming, Series A

    (1989)
  • E.J. Anderson et al.

    Linear Programming in Infinite Dimensional Spaces

    (1987)
  • N.N. Astaf'ev, Regularization of a semi-infinite linear programming problem, in: L.V. Bokut' et al. (Ed.), Siberian...
  • A. Auslender et al.

    On closed convex sets without boundary rays and asymptotes

    Set-Valued Analysis

    (1994)
  • G. Beer

    Topologies on closed and closed convex sets

    (1993)
  • A. Ben-Tal et al.

    Robust convex optimization

    Mathematics of Operations Research

    (1998)
  • B. Betró, An accelerated central cutting plane algorithm for linear semi-infinite programming, 2001,...
  • B. Betró et al.

    Methods for global prior robustness under generalized moment conditions

  • K. Bezdek

    On the illumination of smooth convex bodies

    Archiv der Mathematik

    (1992)
  • M.J. Cánovas, Estabilidad de sistemas de desigualdades lineales en un contexto paramétrico, Ph.D. thesis, Miguel...
  • M.J. Cánovas, M.A. López, J. Parra, Stability of linear inequality systems in a parametric setting, Technical report,...
  • M.J. Cánovas, M.A. López, J. Parra, Stability of the feasible set for linear inequality systems: A carrier index set...
  • M.J. Cánovas et al.

    Stability and well-posedness in linear semi-infinite programming

    SIAM Journal on Optimization

    (1999)
  • M.J. Cánovas et al.

    Solving strategies and well-posedness in linear semi-infinite programming

    Annals of Operations Research

    (2001)
  • A. Charnes et al.

    Duality, Haar programs and finite sequence spaces

    Proceedings of the National Academy of Science

    (1962)
  • A. Charnes et al.

    Duality in semi-infinite programs and some works of Haar and Carathéodory

    Management Science

    (1963)
  • M. Dall'Aglio, On some applications of lsip to probability and statistics, in: M.A. Goberna, M.A. López, (Eds.),...
  • A.L. Dontchev et al.

    Well-posed optimization problems

    (1993)
  • D. Dugosija

    On stability in quasiconvex semi-infinite optimization

    Yugoslav Journal of Operations Research

    (1991)
  • N. Elia et al.

    l1-minimization with magnitude constraints in the frequency domain

    Journal of Optimization Theory and Applications

    (1997)
  • M.D. Fajardo et al.

    Locally Farkas–Minkowski systems in convex semi-infinite programming

    Journal of Optimization Theory and Applications

    (1999)
  • P. Felcyn

    Limimal number of a convex body with a smooth boundary

    Demonstratio Mathematica

    (1988)
  • D. Gale et al.

    Convex functions on convex polytopes

    Mathematica Scandinavica

    (1959)
  • M.A. Goberna et al.

    Geometric fundamentals of the simplex method in semi-infinite programming

    Operations Research Spektrum

    (1988)
  • M.A. Goberna et al.

    Analytical linear inequality systems and optimization

    Journal of Optimization Theory and Applications

    (1999)
  • M.A. Goberna, V. Jornet, M. Rodrı́guez, On the characterization of some families of closed convex sets, Contributions...
  • M.A. Goberna et al.

    A comprehensive survey of linear semi-infinite optimization

  • M.A. Goberna et al.

    Linear semi-infinite optimization

    (1998)
  • M.A. Goberna, M.A. López, S.Y. Wu, Separation by hyperplanes: A linear semi-infinite programming approach, in: M.A....
  • F. Guerra et al.

    On feasible sets defined through Chebyshev approximation

    Mathematical Methods of Operations Research

    (1998)
  • M. Gugat

    Convex semi-infinite parametric programming: Uniform convergence of the optimal value functions of discretized problems

    Journal of Optimization Theory and Applications

    (1999)
  • S.A. Gustafson

    Calculating bounds for a class of Laplace integrals, Parametric optimization and related topics IV

  • S.-A. Gustafson et al.

    On accurate computation of a class of linear functionals

    Journal of Mathematical Systems, Estimation and Control

    (1998)
  • J.-B. Hiriart-Urruty et al.

    Convex Analysis and Minimization Algorithms I

    (1991)
  • A.J. Hoffman

    On approximate solutions of systems of linear inequalities

    Journal of Research of the National Bureau of Standards

    (1952)
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