Strukturált nemlineáris programozási feladatok: elmélet, algoritmusok és alkalmazások = Structured Nonlinear Programming Problems: Theory, Algorithms and Applications

Kutatásunk középpontjában a lineáris optimalizálás területén kifejlesztett pivot- illetve belsőpontos algoritmusok általánosításának a kérdései álltak. Általános lineáris komplementaritási feladatok (LCP) megoldásával foglalkoztunk, amelyeknek számos alkalmazási területe van, mint például a játékelmélet vagy a gazdasági egyensúlyi modellek. Algoritmusaink kidolgozásához nélkülözhetetlen volt a megfelelő elméleti háttér megismerése és szükség esetén annak a célirányos továbbfejlesztése. Ezért foglalkoztunk az LCP-k dualitás elméletével, a témakörhöz kapcsolódó EP-tétellel és a kapcsolódó mátrixosztályok tulajdonságaival. Általános LCP feladatok megoldására alkalmas criss-cross algoritmust fejlesztettünk ki, foglalkoztunk a módszer ciklizálás mentességének a kérdésével (új ciklizálás ellenes szabályokat fogalmaztunk meg és vizsgáltunk). Algoritmusunk hatékonyságát numerikus teszteken mutattuk be. A belsőpontos módszerek legfőbb osztályainak (útkövető-; affin skálázású-; prediktor-korrektor algoritmusok) egy-egy képviselőjét általánosítottuk, általános LCP feladatok EP-megoldásának az előállítására. Algoritmusaink EP-megoldást polinom időben állítanak elő és alkalmasak arra is, hogy általános LCP feladatokat - klasszikus értelemben - oldjanak meg. Foglalkoztunk például szeparábilis konkáv célfüggvényes, lineáris feltételes minimalizálási feladattal és más alkalmazásból érkező bonyolult optimalizálási feladatokkal. | Our main goal was to extend the applicability of pivot and interior point algorithms from linear optimization problem to a wider class of optimization problems. We were dealing with solvability of general linear complementarity problem (G-LCP) that has many interesting application area like game theory or economical equilibrium problems. In our research it was essential to learn and - when it was necessary - to further develop the duality theory of LCPs, EP-theorems and properties of important matrix classes. We developed new variants of criss-cross algorithms for GLCP, introduced new anti-cycling pivot rules and tested its efficiency on practical problems. One member from each main class of interior point (path-following-, affine scaling-, predictor-corrector) algorithms (IPA) has been generalized to GLCP problems. These general IPAs solve the GLCP problem in EP-sense with polynomial running time. Furthermore, these algorithms are appropriate tools for computing solution of GLCP, as well. In the past 5 years, during this research project we worked on other interesting, structured nonlinear programming problems like separable concave minimization problem with linear constraints as well

On the spherical convexity of quadratic functions

In this paper we study the spherical convexity of quadratic functions on spherically convex sets. In particular, conditions characterizing the spherical convexity of quadratic functions on spherical convex sets associated to the positive orthants and Lorentz cones are given

RSP-Based Analysis for Sparsest and Least ℓ1\ell_1-Norm Solutions to Underdetermined Linear Systems

Recently, the worse-case analysis, probabilistic analysis and empirical justification have been employed to address the fundamental question: When does $\ell_1$-minimization find the sparsest solution to an underdetermined linear system? In this paper, a deterministic analysis, rooted in the classic linear programming theory, is carried out to further address this question. We first identify a necessary and sufficient condition for the uniqueness of least $\ell_1$-norm solutions to linear systems. From this condition, we deduce that a sparsest solution coincides with the unique least $\ell_1$-norm solution to a linear system if and only if the so-called \emph{range space property} (RSP) holds at this solution. This yields a broad understanding of the relationship between $\ell_0$- and $\ell_1$-minimization problems. Our analysis indicates that the RSP truly lies at the heart of the relationship between these two problems. Through RSP-based analysis, several important questions in this field can be largely addressed. For instance, how to efficiently interpret the gap between the current theory and the actual numerical performance of $\ell_1$-minimization by a deterministic analysis, and if a linear system has multiple sparsest solutions, when does $\ell_1$-minimization guarantee to find one of them? Moreover, new matrix properties (such as the \emph{RSP of order $K$} and the \emph{Weak-RSP of order $K$}) are introduced in this paper, and a new theory for sparse signal recovery based on the RSP of order $K$ is established

Lattice-like operations and isotone projection sets

By using some lattice-like operations which constitute extensions of ones introduced by M. S. Gowda, R. Sznajder and J. Tao for self-dual cones, a new perspective is gained on the subject of isotonicity of the metric projection onto the closed convex sets. The results of this paper are wide range generalizations of some results of the authors obtained for self-dual cones. The aim of the subsequent investigations is to put into evidence some closed convex sets for which the metric projection is isotonic with respect the order relation which give rise to the above mentioned lattice-like operations. The topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool. For Euclidean sublattices this approach was considered by G. Isac and respectively by H. Nishimura and E. A. Ok.Comment: Proofs of Theorem 1 and Corollary 4 have been corrected. arXiv admin note: substantial text overlap with arXiv:1210.232

Reformulations of mathematical programming problems as linear complementarity problems

A family of complementarity problems are defined as extensions of the well known Linear Complementarity Problem (LCP). These are (i.) Second Linear Complementarity Problem (SLCP) which is an LCP extended by introducing further equality restrictions and unrestricted variables, (ii.) Minimum Linear Complementarity Problem (MLCP) which is an LCP with additional variables not required to be complementary and with a linear objective function which is to be minimized, (iii.) Second Minimum Linear Complementarity Problem (SMLCP) which is an MLCP but the nonnegative restriction on one of each pair of complementary variables is relaxed so that it is allowed to be unrestricted in value. A number of well known mathematical programming problems, namely quadratic programming (convex, nonconvex, pseudoconvex nonconvex), bilinear programming, game theory, zero-one integer programming, the fixed charge problem, absolute value programming, variable separable programming are reformulated as members of this family of four complementarity problems