On Linear Inequality Systems with Smooth Coefficients

A linear inequality system with infinitely many constraints is polynomial [analytical] if its index set is a compact interval of the real line and all its coefficients are polynomial [analytical] functions of the index on this interval. This paper provides sufficient conditions for a given closed convex set to be the solution set of a certain polynomial or at least analytical system.The authors are indebted to Dr. J. M. Almira for valuable comments and suggestions.     

Analytical Linear Inequality Systems and Optimization

In many interesting semi-infinite programming problems, all the constraints are linear inequalities whose coefficients are analytical functions of a one-dimensional parameter. This paper shows that significant geometrical information on the feasible set of these problems can be obtained directly from the given coefficient functions. One of these geometrical properties gives rise to a general purification scheme for linear semi-infinite programs equipped with so-called analytical constraint systems. It is also shown that the solution sets of such kind of consistent systems form a transition class between polyhedral convex sets and closed convex sets in the Euclidean space of the unknowns.     

Maximum Feasibility Problem for Continuous Linear Inequalities with Applications to Fuzzy Linear Programming

Systems of linear inequalities are important tools to formulate optimization problems. However, the feasibility of the whole system was often presumed true in most models. Even if an infeasible system could be detected, it is in general not easy to tell which part of the system caused it. This motivates the study of continuous linear inequalities, given no information whether it is feasible or not, what is the largest possible portion of the system that can be remained in consistency? We first propose a bisection-based algorithm which comes with an auxiliary program to answer the question. For further accelerating the algorithm, several novel concepts, one called “constraint weighting” and the other called “shooting technique”, are introduced to explore intrinsic problem structures. This new scheme eventually replaces the bisection method and its validity can be justified via a solid probabilistic analysis. Numerical examples and applications to fuzzy inequalities are reported to illustrate the robustness of our algorithm.     

Relaxed Cutting Plane Method for Solving Linear Semi-Infinite Programming Problems

One of the major computational tasks of using the traditional cutting plane approach to solve linear semi-infinite programming problems lies in finding a global optimizer of a nonlinear and nonconvex program. This paper generalizes the Gustafson and Kortanek scheme to relax this requirement. In each iteration, the proposed method chooses a point at which the infinite constraints are violated to a degree, rather than a point at which the violations are maximized. A convergence proof of the proposed scheme is provided. Some computational results are included. An explicit algorithm which allows the unnecessary constraints to be dropped in each iteration is also introduced to reduce the size of computed programs.     

Separating the solution sets of analytical and polynomial systems

A linear inequality system with infinitely many constraints is polynomial (analytical) if its index set is a compact interval of the real line and all its coefficients are polynomial (analytical, respectively) functions of the index on this interval. This paper provides an example of analytical system whose solution set cannot be the solution set of any polynomial system. Research supported by DGES of Spain and FEDER of UE, Grant BFM2002-04114-C02-01. Research supported by CONACyT of Mexico, Grant 130036. Research partially supported by CONACyT of Mexico, Grant 44003.     

Uniform saturation in linear inequality systems

Redundant constraints in linear inequality systems can be characterized as those inequalities that can be removed from an arbitrary linear optimization problem posed on its solution set without modifying its value and its optimal set. A constraint is saturated in a given linear optimization problem when it is binding at the optimal set. Saturation is a property related with the preservation of the value and the optimal set under the elimination of the given constraint, phenomena which can be seen as weaker forms of excess information in linear optimization problems. We say that an inequality of a given linear inequality system is uniformly saturated when it is saturated for any solvable linear optimization problem posed on its solution set. This paper characterizes the uniform saturated inequalities and other related classes of inequalities. This work was supported by the MCYT of Spain and FEDER of UE, Grant BFM2002-04114-C02-01.     

Linear programming with variable matrix entries

We consider linear programming (continuous or integer) where some matrix entries are decision parameters. If the variables are nonnegative the problem can be easily solved in two phases. It is shown that direct costs on the matrix entries make the problem NP-hard. Finally, a strong duality result is provided.     

On the stability of linear systems with an exact constraint set

This paper deals with the stability of the intersection of a given set with the solution, , of a given linear system whose coefficients can be arbitrarily perturbed. In the optimization context, the fixed constraint set X can be the solution set of the (possibly nonlinear) system formed by all the exact constraints (e.g., the sign constraints), a discrete subset of (as or { 0,1} ⁿ , as it happens in integer or Boolean programming) as well as the intersection of both kind of sets. Conditions are given for the intersection to remain nonempty (or empty) under sufficiently small perturbations of the data. Research supported by Fondecyt Grant 1020(7020)-646. Research supported by DGES and FEDER, Grant BFM2002-04114-C02-01     

线性规划在回归分析中的应用

线性回归中,针对最小二乘法的两个替代准则一绝对离差和最小准则以及最大绝对离差最小准则,利用线性规划技术建立回归预测模型。实用分析表明,线性规划模型具有较好的预测效果,有郊地消除了统计数据中异常值对回归方程的影响。     

A Dual Parametrization Method for Convex Semi-Infinite Programming

We formulate convex semi-infinite programming problems in a functional analytic setting and derive optimality conditions and several duality results, based on which we develop a computational framework for solving convex semi-infinite programs.     

Posynomial geometric programming as a special case of semi-infinite linear programming

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.     

On the Stability of the Feasible Set in Linear Optimization

This paper deals with the stability of two families of linear optimization problems, each one formed by the dual problems to the members of the other family. We characterize the problems of these families that are stable in the sense that they remain consistent (inconsistent) under sufficiently small arbitrary perturbations of all the data. This characterization is established in terms of the lower semicontinuity property of the feasible set mapping and the boundedness of the optimal set of the corresponding coupled problem. Other continuity properties of the feasible set mapping are also derived. This stability theory extends some well-known theorems of Williams and Robinson on the stability of ordinary linear programming problems to linear optimization problems with infinitely many variables or constraints.     

Locally Farkas-Minkowski linear inequality systems

This paper introduces thelocally Farkas-Minkowski (LFM) linear inequality systems in a finite dimensional Euclidean space. These systems are those ones that satisfy that any consequence of the system that is active at some solution point is also a consequence of some finite subsystem. This class includes the Farkas-Minkowski systems and verifies most of the properties that these systems possess. Moreover, it contains the locally polyhedral systems, which are the natural external representation of quasi-polyhedral sets. TheLFM systems appear to be the natural external representation of closed convex sets. A characterization based on their properties under the union of systems is provided. In linear semi-infinite programming, theLFM property is the more general constraint qualification such that the Karush-Kuhn-Tucker condition characterizes the optimal points. Furthermore, the pair of Haar dual problems has no duality gap.     

(A,B)-Invariant Polyhedral Sets of Linear Discrete-Time Systems

The problem of confining the trajectory of a linear discrete-time system in a given polyhedral domain is addressed through the concept of (A, B)-invariance. First, an explicit characterization of (A, B)-invariance of convex polyhedra is proposed. Such characterization amounts to necessary and sufficient conditions in the form of linear matrix relations and presents two major advantages compared to the ones found in the literature: it applies to any convex polyhedron and does not require the computation of vertices. Such advantages are felt particularly in the computation of the supremal (A, B)-invariant set included in a given polyhedron, for which a numerical method is proposed. The problem of computing a control law which forces the system trajectories to evolve inside an (A, B)-invariant polyhedron is treated as well. Finally, the (A, B)-invariance relations are generalized to persistently disturbed systems.     

Hedging Interest Rate Risk by Optimization in Banach Spaces

This paper addresses the hedging of bond portfolios interest rate risk by drawing on the classical one-period no-arbitrage approach of financial economics. Under quite weak assumptions, several maximin portfolios are introduced by means of semi-infinite mathematical programming problems. These problems involve several Banach spaces; consequently, infinite-dimensional versions of classical algorithms are required. Furthermore, the corresponding solutions satisfy a saddle-point condition illustrating how they may provide appropriate hedging with respect to the interest rate risk. Communicated by D. G. Luenberger This research was partially supported by the Comunidad Autonoma de Madrid (Spain), Grant 06/HSE/0150/2004, and by MEyC (Spain), Grants BEC2000-1388-C04-03 and BEC2000-0167. The authors thank the unknown reviewer whose suggestions have led to significant improvements of the paper.     

Stability of the primal-dual partition in linear semi-infinite programming

We analyse the primal-dual states in linear semi-infinite programming (LSIP), where we consider the primal problem and the so called Haar's dual problem. Any linear programming problem and its dual can be classified as bounded, unbounded or inconsistent, giving rise to nine possible primal-dual states, which are reduced to six by the weak duality property. Recently, Goberna and Todorov have studied this partition and its stability in continuous LSIP in a series of papers [M.A. Goberna and M.I. Todorov, Primal, dual and primal-dual partitions in continuous linear semi-infinite programming, Optimization 56 (2007), pp. 617–628; M.A. Goberna and M.I. Todorov, Generic primal-dual solvability in continuous linear semi-infinite programming, Optimization 57 (2008), pp. 239–248]. In this article we consider the general case, with no continuity assumptions, discussing the maintenance of the primal-dual state of the problem by allowing small perturbations of the data. We characterize the stability of all of the six possible primal-dual states through necessary and sufficient conditions which depend on the data, and can be easily checked, showing some differences with the continuous case. These conditions involve the strong Slater constraint qualification, and some distinguished convex sets associated to the data.     

Sensitivity analysis in linear semi-infinite programming: Perturbing cost and right-hand-side coefficients

This paper analyzes the effect on the optimal value of a given linear semi-infinite programming problem of the kind of perturbations which more frequently arise in practical applications: those which affect the objective function and the right-hand-side coefficients of the constraints. In particular, we give formulae which express the exact value of a perturbed problem as a linear function of the perturbation.     

On the Stability of the Boundary of the Feasible Set in Linear Optimization

This paper analizes the relationship between the stability properties of the closed convex sets in finite dimensions and the stability properties of their corresponding boundaries. We consider a given closed convex set represented by a certain linear inequality system whose coefficients can be arbitrarily perturbed, and we measure the size of these perturbations by means of the pseudometric of the uniform convergence. It is shown that the feasible set mapping is Berge lower semicontinuous at if and only if the boundary mapping satisfies the same property. Moreover, if the boundary mapping is semicontinuous in any sense (lower or upper; Berge or Hausdorff) at , then it is also closed at . All the mentioned stability properties are equivalent when the feasible set is a convex body.     

Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization

We present the convergence analysis of the inexact infeasible path-following (IIPF) interior-point algorithm. In this algorithm, the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix. The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of the interior-point method for this specific inexact case. We present the convergence analysis of the inexact infeasible path-following (IIPF) algorithm, prove the global convergence of this method and provide complexity analysis. Communicated by Y. Zhang.