Linear semi-infinite programming theory: An updated survey
Introduction
Every LP problem can be formulated in standard format aswhere the unknowns are λt, t=1,2,…,m, and the data are , , and , t=1,2,…,m. Its corresponding dual problem can be written aswith 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, t∈T, to be non-zero. In other words, the space of the variables of the first problem, , is replaced with the linear space of all the functions such that λt=0 for all t∈T except maybe for a finite number of indices, the supporting set of λ, that we denote by . This linear space is denoted by (the so-called space of generalized finite sequences), and its positive cone by . Thus, we associate with the given data, which form the triple (P)=(c,a.,b.), with , , and , the pair of problemsandThe constraint system of (P), σ={at′x⩾bt,t∈T} (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 a′x⩾b is a consequence of σ if it is satisfied for all x∈F. 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 , whose zero-vector, open unit ball and unit sphere will be denoted by 0n, Bn and Sn, respectively. The canonical basis of shall be represented by e1,…,en. For any set X≠∅, let us denote by spanX, , , 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 , , and the closure, the boundary, the relative boundary and the relative interior of X (relative means with respect to the topology of ). We denote by the dimension of X (i.e., the dimension of ), by the dimension of , and by O+X the cone of recession directions of X. A vector is a feasible direction at x∈X if there exists ε>0 such that x+εy∈X. 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 x∈F, where 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 instead of the nominal one, . In order to measure the size of the perturbations, we restrict to the present parameter space (identified with the extended distance d introduced in the previous section,
Convex semi-infinite programming
In the optimality theory for the ordinary convex optimization problemwhere f and gt, t∈T, are real convex finite-valued functions on , the basic constraint qualification (BCQ, in short) plays an important role. We say that the constraints system satisfies the BCQ ifwhere NF(x) is the normal cone to the feasible set F at the point x; i.e.,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)
- et al.
Locally polyhedral linear semi-infinite systems
Linear Algebra and its Applications
(1998) - et al.
Linear programming with fuzzy coefficients in constraints
Computers and Mathematics with Applications
(1999) - et al.
Logarithmic barrier decomposition methods for semi-infinite programming
International Transactions in Operations Research
(1997) - et al.
On the numerical treatment of linearly constrained semi-infinite optimization problems
European Journal of Operations Research
(2000) - et al.
Complexity analysis of logaritmic barrier decomposition for semi-infinite linear programming
Applied Numerical Mathematics
(1999) - et al.
Quasi-polyhedral sets in semi-infinite linear inequality systems
Linear Algebra and its Applications
(1997) Pointwise versions of the maximum theorem with applications in optimization
Applied Mathematics Letters
(1990)- et al.
A perturbation method for solving linear semi-infinite programming problems
Computers and Mathematics with Applications
(1999) - et al.
Solving convex programs with infinitely many linear constraints by a relaxed cutting plane method
Computers and Mathematics with Applications
(1999) - L. Abbe, Two logaritmic barrier methods for convex semi-infinite problems, in: M.A. Goberna, M.A. López, (Eds.),...
A dynamic model for component testing
Naval Research Logistics
Strong duality for inexact linear programming problems
Optimization
Simplex-like trajectories on quasi-polyhedral convex sets
Mathematics of Operations Research
An extension of the simplex algorithm for semi-infinite linear programming
Mathematical Programming, Series A
Linear Programming in Infinite Dimensional Spaces
On closed convex sets without boundary rays and asymptotes
Set-Valued Analysis
Topologies on closed and closed convex sets
Robust convex optimization
Mathematics of Operations Research
Methods for global prior robustness under generalized moment conditions
On the illumination of smooth convex bodies
Archiv der Mathematik
Stability and well-posedness in linear semi-infinite programming
SIAM Journal on Optimization
Solving strategies and well-posedness in linear semi-infinite programming
Annals of Operations Research
Duality, Haar programs and finite sequence spaces
Proceedings of the National Academy of Science
Duality in semi-infinite programs and some works of Haar and Carathéodory
Management Science
Well-posed optimization problems
On stability in quasiconvex semi-infinite optimization
Yugoslav Journal of Operations Research
l1-minimization with magnitude constraints in the frequency domain
Journal of Optimization Theory and Applications
Locally Farkas–Minkowski systems in convex semi-infinite programming
Journal of Optimization Theory and Applications
Limimal number of a convex body with a smooth boundary
Demonstratio Mathematica
Convex functions on convex polytopes
Mathematica Scandinavica
Geometric fundamentals of the simplex method in semi-infinite programming
Operations Research Spektrum
Analytical linear inequality systems and optimization
Journal of Optimization Theory and Applications
A comprehensive survey of linear semi-infinite optimization
Linear semi-infinite optimization
On feasible sets defined through Chebyshev approximation
Mathematical Methods of Operations Research
Convex semi-infinite parametric programming: Uniform convergence of the optimal value functions of discretized problems
Journal of Optimization Theory and Applications
Calculating bounds for a class of Laplace integrals, Parametric optimization and related topics IV
On accurate computation of a class of linear functionals
Journal of Mathematical Systems, Estimation and Control
Convex Analysis and Minimization Algorithms I
On approximate solutions of systems of linear inequalities
Journal of Research of the National Bureau of Standards
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