Lineáris optimalizálás : elmélete és belsőpontos algoritmusai

Interior point methods for linear programming release the result of a long process. Today's knowledge, the first notable result was coined Frisch, who in 1955 gave a lecture at a seminar in the University of Oslo in the econometric seminar logarithmic barrier method of linear programming applicability. The method multiple algorithm called it, which was published in 1957. Another result that was almost unnoticed, Diki coined, and in 1967 was published. Diki introduced the ellipsoid named after him, which could help you to approach and approach to solve linear programming problems with a special structure. you define primal, affine scaling interior point algorithms using the method again

The s-monotone index selection rules for pivot algorithms of linear programming

In this paper we introduce the concept of s-monotone index selection rule for linear programming problems. We show that several known anti-cycling pivot rules like the minimal index, Last-In–First-Out and the most-often-selected-variable pivot rules are s-monotone index selection rules. Furthermore, we show a possible way to define new s-monotone pivot rules. We prove that several known algorithms like the primal (dual) simplex, MBU-simplex algorithms and criss-cross algorithm with s-monotone pivot rules are finite methods. We implemented primal simplex and primal MBU-simplex algorithms, in MATLAB, using three s-monotone index selection rules, the minimal-index, the Last-In–First-Out (LIFO) and the Most-Often-Selected-Variable (MOSV) index selection rule. Numerical results demonstrate the viability of the above listed s-monotone index selection rules in the framework of pivot algorithms

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

Predictor-corrector interior-point algorithm for sufficient linear complementarity problems based on a new type of algebraic equivalent transformation technique

We propose a new predictor-corrector (PC) interior-point algorithm (IPA) for solving linear complementarity problem (LCP) with P_* (κ)-matrices. The introduced IPA uses a new type of algebraic equivalent transformation (AET) on the centering equations of the system defining the central path. The new technique was introduced by Darvay et al. [21] for linear optimization. The search direction discussed in this paper can be derived from positive-asymptotic kernel function using the function φ(t)=t^2 in the new type of AET. We prove that the IPA has O(1+4κ)√n log⁡〖(3nμ^0)/ε〗 iteration complexity, where κ is an upper bound of the handicap of the input matrix. To the best of our knowledge, this is the first PC IPA for P_* (κ)-LCPs which is based on this search direction

Unified approach of primal-dual interior-point algorithms for a new class of AET functions

We propose new short-step interior-point algorithms (IPAs) for solving P_* (κ)-linear complementarity problems (LCPs). In order to define the search directions we use the algebraic equivalent transformation technique (AET) of the system which characterizes the central path. A novelty of the paper is that we introduce a new class of AET functions. We present the complexity analysis of the IPAs that use this general class of functions in the AET technique. Furthermore, we also deal with a special case, namely φ(t)=t^2-t+√t. This function differs from the ones used in the literature in the sense that it has inflection point. It does not belong to the class of concave functions determined by Haddou et al. Furthermore, the kernel function corresponding to this AET function is neither eligible nor self-regular kernel function. We prove that the IPAs using any member φ of this new class of AET functions have polynomial iteration complexity in the size of the problem, bit length of the integral data and in the parameter κ. Beside this, we also provide numerical results that show the efficiency of the introduced methods

Unified Approach of Interior-Point Algorithms for P_*(\kappa )-LCPs Using a New Class of Algebraically Equivalent Transformations

We propose new short-step interior-point algorithms (IPAs) for solving P_*(\kappa ) P ∗ ( κ ) -linear complementarity problems (LCPs). In order to define the search directions, we use the algebraic equivalent transformation (AET) technique of the system describing the central path. A novelty of the paper is that we introduce a whole, new class of AET functions for which a unified complexity analysis of the IPAs is presented. This class of functions differs from the ones used in the literature for determining search directions, like the class of concave functions determined by Haddou, Migot and Omer, self-regular functions, eligible kernel and self-concordant functions. We prove that the IPAs using any member \varphi φ of the new class of AET functions have polynomial iteration complexity in the size of the problem, in starting point’s duality gap, in the accuracy parameter and in the parameter \kappa κ

New predictor-corrector interior-point algorithm with AET function having inflection point

In this paper we introduce a new predictor-corrector interior-point algorithm for solving P_* (κ)-linear complementarity problems. For the determination of search directions we use the algebraically equivalent transformation (AET) technique. In this method we apply the function φ(t)=t^2-t+√t which has inflection point. It is interesting that the kernel corresponding to this AET function is neither self-regular, nor eligible. We present the complexity analysis of the proposed interior-point algorithm and we show that it's iteration bound matches the best known iteration bound for this type of PC IPAs given in the literature. It should be mentioned that usually the iteration bound is given for a fixed update and proximity parameter. In this paper we provide a set of parameters for which the PC IPA is well defined. Moreover, we also show the efficiency of the algorithm by providing numerical results

Predictor-corrector interior-point algorithm based on a new search direction working in a wide neighbourhood of the central path

We introduce a new predictor-corrector interior-point algorithm for solving P_*(κ)-linear complementarity problems which works in a wide neighbourhood of the central path. We use the technique of algebraic equivalent transformation of the centering equations of the central path system. In this technique, we apply the function φ(t)=√t in order to obtain the new search directions. We define the new wide neighbourhood D_φ. In this way, we obtain the first interior-point algorithm, where not only the central path system is transformed, but the definition of the neighbourhood is also modified taking into consideration the algebraic equivalent transformation technique. This gives a new direction in the research of interior-point methods. We prove that the IPA has O((1+κ)n log⁡((〖〖(x〗^0)〗^T s^0)/ϵ) ) iteration complexity. Furtermore, we show the efficiency of the proposed predictor-corrector interior-point method by providing numerical results. Up to our best knowledge, this is the first predictor-corrector interior-point algorithm which works in the D_φ neighbourhood using φ(t)=√t