Global optimality conditions and optimization methods for constrained polynomial programming problems

The general constrained polynomial programming problem (GPP) is considered in this paper. Problem (GPP) has a broad range of applications and is proved to be NP-hard. Necessary global optimality conditions for problem (GPP) are established. Then, a new local optimization method for this problem is proposed by exploiting these necessary global optimality conditions. A global optimization method is proposed for this problem by combining this local optimization method together with an auxiliary function. Some numerical examples are also given to illustrate that these approaches are very efficient. (C) 2015 Elsevier Inc. All rights reserved

Transposition theorems and qualification-free optimality conditions

New theorems of the alternative for polynomial constraints (based on the Positivstellensatz from real algebraic geometry) and for linear constraints (generalizing the transposition theorems of Motzkin and Tucker) are proved. Based on these, two Karush-John optimality conditions – holding without any constraint qualification – are proved for single- or multi-objective constrained optimization problems. The first condition applies to polynomial optimization problems only, and gives for the first time necessary and sufficient global optimality conditions for polynomial problems. The second condition applies to smooth local optimization problems and strengthens known local conditions. If some linear or concave constraints are present, the new version reduces the number of constraints for which a constraint qualification is needed to get the Kuhn-Tucker conditions

Global optimality conditions and optimization methods for polynomial programming problems and their applications

The polynomial programming problem which has a polynomial objective function, either with no constraints or with polynomial constraints occurs frequently in engineering design, investment science, control theory, network distribution, signal processing and locationallocation contexts. Moreover, the polynomial programming problem is known to be Nondeterministic Polynomial-time hard (NP-hard). The polynomial programming problem has attracted a lot of attention, including quadratic, cubic, homogenous or normal quartic programming problems as special cases. Existing methods for solving polynomial programming problems include algebraic methods and various convex relaxation methods. Especially, among these methods, semidefinite programming (SDP) and sum of squares (SOS) relaxations are very popular. Theoretically, SDP and SOS relaxation methods are very powerful and successful in solving the general polynomial programming problem with a compact feasible region. However, the solvability in practice depends on the size or the degree of the polynomial programming problem and the required accuracy. Hence, solving large scale SDP problems still remains a computational challenge. It is well-known that traditional local optimization methods are designed based on necessary local optimality conditions, i.e., Karush-Kuhn-Tucker (KKT) conditions. Motivated by this, some researchers proposed a necessary global optimality condition for a quadratic programming problem and designed a new local optimization method according to the necessary global optimality condition. In this thesis, we try to apply this idea to cubic and quatic programming problems, and further to general unconstrained and constrained polynomial programming problems. For these polynomial programming problems, we will investigate necessary global optimality conditions and design new local optimization methods according to these conditions. These necessary global optimality conditions are generally stronger than KKT conditions. Hence, the obtained new local minimizers by using the new local optimization methods may improve some KKT points. Our ultimate aim is to design global optimization methods for these polynomial programming problems. We notice that the filled function method is one of the well-known and practical auxiliary function methods used to achieve a global minimizer. In this thesis, we design global optimization methods by combining the new proposed local optimization methods and some auxiliary functions. The numerical examples illustrate the efficiency and stability of the optimization methods. Finally, we discuss some applications for solving some sensor network localization problems and systems of polynomial equations. It is worth mentioning that we apply the idea and the results for polynomial programming problems to nonlinear programming problems (NLP). We provide an optimality condition and design new local optimization methods according to the optimality condition and design global optimization methods for the problem (NLP) by combining the new local optimization methods and an auxiliary function. In order to test the performance of the global optimization methods, we compare them with two other heuristic methods. The results demonstrate our methods outperform the two other algorithms.Doctor of Philosoph

A feasible second order bundle algorithm for nonsmooth, nonconvex optimization problems with inequality constraints and its application to certificates of infeasibility

Diese Arbeit erweitert den SQP-Zugang des Bundle-Newton-Verfahrens für nichtglatte, unrestringierte Optimierungsprobleme zu einem zulässigen Bundle-Algorithmus zweiter Ordnung für nichtglatte, nichtkonvexe Optimierungsprobleme mit Ungleichungsnebenbedingungen. An Stelle der Verwendung einer Straffunktion oder eines Filters oder einer Improvement-Funktion zur Behandlung der Nebenbedingungen, wird die Suchrichtung durch Lösen eines konvexen quadratischen Optimierungsproblems mit quadratischen Nebenbedingungen bestimmt, um gute Iterationspunkte zu erhalten. Außerdem untersuchen wir einige Varianten des Suchrichtungsproblems, wir geben eine numerische Rechtfertigung für die Anwendbarkeit des vorgestellten Zugangs, indem wir die Effektivität von verschiedener Lösungssoftware für die Berechnung der Suchrichtung vergleichen, und wir weisen die globale Konvergenz der Methode unter bestimmten Voraussetzungen nach. Weiters stellen wir eine wichtige Anwendung der nichtglatten Optimierung für Zulässigkeitsprobleme vor: Dazu führen wir ein Unzulässigkeitszertifikat ein, welches das Auffinden von Ausschlussboxen durch Lösen eines nichtglatten Optimierungsproblems mit linearen Nebenbedingungen ermöglicht. Zusätzlich kann dieses Zertifikat verwendet werden, um eine Ausschlussbox durch Lösen eines nichtglatten Optimierungsproblems mit nichtlinearen Nebenbedingungen zu vergrößern. Schließlich besprechen wir noch die im Vergleich zu anderer Lösungssoftware guten Testergebnisse von unserem Bundle-Algorithmus zweiter Ordnung für einige Hock-Schittkowski-Beispiele, für Beispiele die im Zusammenhang mit der Auffindung von Ausschlussboxen in Zulässigkeitsproblemen auftreten und für höher dimensionale stückweise quadratische Beispiele.This thesis extends the SQP-approach of the well-known bundle-Newton method for nonsmooth unconstrained minimization to a feasible second order bundle algorithm for nonsmooth, nonconvex optimization problems with inequality constraints: Instead of using a penalty function or a filter or an improvement function to deal with the presence of constraints, the search direction is determined by solving a convex quadratically constrained quadratic program to obtain good iteration points. Moreover, we investigate certain versions of the search direction problem, we justify the applicability of this approach numerically by using different solvers for the computation of the search direction and we show global convergence of the method under certain assumptions. Furthermore, we present an important application of nonsmooth optimization to constraint satisfaction problems: We introduce a certificate of infeasibility for finding exclusion boxes by solving a linearly constrained nonsmooth optimization problem. Additionally, the constructed certificate can be used to enlarge an exclusion box by solving a nonlinearly constrained nonsmooth optimization problem. Finally, the good performance of the second order bundle algorithm is demonstrated by comparison with test results of other solvers on examples of the Hock-Schittkowski collection, on custom examples that arise in the context of finding exclusion boxes for constraint satisfaction problems, and on higher dimensional piecewise quadratic examples