Milan Hladík

Optimal value range in interval linear programming

We deal with the linear programming problem in which input data can vary in some given real compact intervals. The aim is to compute the exact range of the optimal value function. We present a general approach to the situation the feasible set is described by an arbitrary linear interval system. Moreover, certain dependencies between the constraint matrix coefficients can be involved. As long as we are able to characterize the primal and dual solution set (the set of all possible primal and dual feasible solutions, respectively), the bounds of the objective function result from two nonlinear programming problems. We demonstrate our approach on various cases of the interval linear programming problem (with and without dependencies).

Bounds on Real Eigenvalues and Singular Values of Interval Matrices

We study bounds on real eigenvalues of interval matrices, and our aim is to develop fast computable formulae that produce as-sharp-as-possible bounds. We consider two cases: general and symmetric interval matrices. We focus on the latter case, since on the one hand such interval matrices have many applications in mechanics and engineering, and on the other hand many results from classical matrix analysis could be applied to them. We also provide bounds for the singular values of (generally nonsquare) interval matrices. Finally, we illustrate and compare the various approaches by a series of examples.

Generalized linear fractional programming under interval uncertainty

Data in many real-life engineering and economical problems suffer from inexactness. Herein we assume that we are given some intervals in which the data can simultaneously and independently perturb. We consider a generalized linear fractional programming problem with interval data and present an efficient method for computing the range of optimal values. The method reduces the problem to solving from two to four real-valued generalized linear fractional programs, which can be computed in polynomial time using an appropriate interior point method solver. We consider also the inverse problem: How much can data of a real generalized linear fractional program vary such that the optimal values do not exceed some prescribed bounds. We propose a method for calculating (often the largest possible) ranges of admissible variations; it needs to solve only two real-valued generalized linear fractional programs. We illustrate the approach on a simple von Neumann economic growth model.

How to determine basis stability in interval linear programming

Interval linear programming (ILP) was introduced in order to deal with linear programming problems with uncertainties that are modelled by ranges of admissible values. Basic tasks in ILP such as calculating the optimal value bounds or set of all possible solutions may be computationally very expensive. However, if some basis stability criterion holds true then the problems becomes much more easy to solve. In this paper, we propose a method for testing basis stability. Even though the method is exponential in the worst case (not surprisingly due to NP-hardness of the problem), it is fast in many cases.

Enclosures for the solution set of parametric interval linear systems

Abstract We investigate parametric interval linear systems of equations. The main result is a generalization of the Bauer–Skeel and the Hansen–Bliek–Rohn bounds for this case, comparing and refinement of both. We show that the latter bounds are not provable better, and that they are also sometimes too pessimistic. The presented form of both methods is suitable for combining them into one to get a more efficient algorithm. Some numerical experiments are carried out to illustrate performances of the methods.

Outer enclosures to the parametric AE solution set

We consider systems of linear equations, where the elements of the matrix and of the right-hand side vector are linear functions of interval parameters. We study parametric AE solution sets, which are defined by universally and existentially quantified parameters, and the former precede the latter. Based on a recently obtained explicit description of such solution sets, we present three approaches for obtaining outer estimations of parametric AE solution sets. The first approach intersects inclusions of parametric united solution sets for all combinations of the end-points of the universally quantified parameters. Polynomially computable outer bounds for parametric AE solution sets are obtained by parametric AE generalization of a single-step Bauer–Skeel method. In the special case of parametric tolerable solution sets, we derive an enclosure based on linear programming approach; this enclosure is optimal under some assumption. The application of these approaches to parametric tolerable and controllable solution sets is discussed. Numerical examples accompanied by graphic representations illustrate the solution sets and properties of the methods.

A filtering method for the interval eigenvalue problem

We consider the general problem of computing intervals that contain the real eigenvalues of interval matrices. Given an outer approximation (superset) of the real eigenvalue set of an interval matrix, we propose a filtering method that iteratively improves the approximation. Even though our method is based on a sufficient regularity condition, it is very efficient in practice and our experimental results suggest that it improves, in general, significantly the initial outer approximation. The proposed method works for general, as well as for symmetric interval matrices.

Complexity of necessary efficiency in interval linear programming and multiobjective linear programming

We present some complexity results on checking necessary efficiency in interval multiobjective linear programming. Supposing that objective function coefficients perturb within prescribed intervals, a feasible point x* is called necessarily efficient if it is efficient for all instances of interval data. We show that the problem of checking necessary efficiency is co-NP-complete even for the case of only one objective. Provided that x* is a non-degenerate basic solution, the problem is polynomially solvable for one objective, but remains co-NP-hard in the general case. Some open problems are mentioned at the end of the paper.

Interval regression by tolerance analysis approach

In interval linear regression analysis, we are given crisp or interval data and we are to determine appropriate interval regression parameters. There are various methods for interval regression; many of them possess the property that while some of the resulting interval regression parameters are very wide, the other parameters are crisp. This drawback is the main limiting factor for such methods and much effort has been devoted to overcoming it. We propose a method motivated by tolerance analysis in linear systems. Our method yields intervals for regression parameters the widths of which are proportional to an in-advance given vector of parameters. Moreover, the method is computationally very cheap, and provides a natural measure of quality of a model. First we formulate the method for the basic model of crisp input-crisp output data and then extend it to crisp input-interval output and interval input-interval output models. For the interval-valued cases we study several formulations of the solution concept: possibility, strong possibility, weak possibility, necessity. Here, strong possibility is a new concept proposed as a natural counterpart to the remaining ones. We prove that the method provides optimal interval parameters meeting centrality and proportionality requirements. We also show that the method provides interval regression parameters satisfying various versions of Tanaka-Lees inclusion property. We also derive a form of a complementarity theorem for the weak possibility and necessity solution concepts. Since practical problems may be affected by outlying observations we show that our approach is easily adapted to deal with them. We illustrate the theory by examples.

Bounds on eigenvalues of real and complex interval matrices

We present a cheap and tight formula for bounding real and imaginary parts of eigenvalues of real or complex interval matrices. It outperforms the classical formulae not only for the complex case but also for the real case. In particular, it generalizes and improves the results by Rohn (1998) [5] and Hertz (2009) [19]. The main idea behind is to reduce the problem to enclosing eigenvalues of symmetric interval matrices, for which diverse methods can be utilized. The result helps in analysing stability of uncertain dynamical systems since the formula gives sufficient conditions for testing Schur and Hurwitz stability of interval matrices. It may also serve as a starting point for some iteration methods.