An algorithm for piece-wise indefinite quadratic programming problem

An indefinite quadratic programming problem is a mathematical programming problem which is a product of two linear factors. In this paper, the piecewise indefinite quadratic programming problem (PIQPP) is considered. Here, the objective function is a product of two continuous piecewise linear functions defined on a non-empty and compact feasible region. In the present paper, the optimality criterion is derived and explained in order to solve PIQPP. While solving PIQPP, we will come across certain variables which will not satisfy the optimality condition. For these variables, cases have been elaborated so as to move from one basic feasible solution to another till we reach the optimality. An algorithmic approach is proposed and discussed for the PIQPP problem. A numerical example is presented to decipher the tendered method

Making Indefinite Kernel Learning Practical

In this paper we embed evolutionary computation into statistical learning theory. First, we outline the connection between large margin optimization and statistical learning and see why this paradigm is successful for many pattern recognition problems. We then embed evolutionary computation into the most prominent representative of this class of learning methods, namely into Support Vector Machines (SVM). In contrast to former applications of evolutionary algorithms to SVM we do not only optimize the method or kernel parameters. We rather use evolution strategies in order to directly solve the posed constrained optimization problem. Transforming the problem into the Wolfe dual reduces the total runtime and allows the usage of kernel functions just as for traditional SVM. We will show that evolutionary SVM are at least as accurate as their quadratic programming counterparts on eight real-world benchmark data sets in terms of generalization performance. They always outperform traditional approaches in terms of the original optimization problem. Additionally, the proposed algorithm is more generic than existing traditional solutions since it will also work for non-positive semidefinite or indefinite kernel functions. The evolutionary SVM variants frequently outperform their quadratic programming competitors in cases where such an indefinite Kernel function is used. --

Properties Of Indefinite Matrix Constraints For Linear Programming In Optimal Solution

Finding the optimum solution in engineering and science is a common problem where one wishes to get the objective under certain constraints.This situation is also a typical issue in manufacturing industries where maximum profit and minimum cost are a common objective under certain constraints on the available resources.One approach to solve optimization is to use formulation problem in linear form and subjects to linear constraints,the problem can be deliberated as linear programming problem.The linear constraints can be in a form of a matrix.There are limited researches that discuss the effect of the properties of matrix constraint to the solution.In fact,the matrix constraint has significant influence to the existent of the optimal solution to the optimization problem.This research focused on the investigation of characteristics of non-symmetric indefinite square matrices of linear programming problems which represent the constraints of linear programming problems.The non-symmetric indefinite square matrices are generated randomly by the MATLAB simulation software and its indefinite properties are verified through the principal minor test,quadratic form test and eigenvalues test.The solutions of the primal and dual linear programming problem are simulated and discussed.Optimization software,LINGO,is used to validate the solutions to assure the reliability of the simulated solutions in the MATLAB software.Based on the simulation results,some of the non-symmetric indefinite random matrices found duality gap and those matrices could not provide optimal solution to the problem.Whereas,some indefinite matrices with certain characteristics could achieve optimal solution and no duality gap presented.An indefinite random matrix with all positive off-diagonal entries and the determinant of leading principal minors with positive sign at odd orders and negative sign at even orders surely deliver the optimal solution to the linear programming problems.This research may contribute to the advancement of linear programming solution particularly when the constraints form an indefinite matrix

Study on Stochastic Linear Quadratic Optimal Control with Quadratic and Mixed Terminal State Constraints

This paper studies the indefinite stochastic LQ control problem with quadratic and mixed terminal state equality constraints, which can be transformed into a mathematical programming problem. By means of the Lagrangian multiplier theorem and Riesz representation theorem, the main result given in this paper is the necessary condition for indefinite stochastic LQ control with quadratic and mixed terminal equality constraints. The result shows that the different terminal state constraints will cause the endpoint condition of the differential Riccati equation to be changed. It coincides with the indefinite stochastic LQ problem with linear terminal state constraint, so the result given in this paper can be viewed as the extension of the indefinite stochastic LQ problem with the linear terminal state equality constraint. In order to guarantee the existence and the uniqueness of the linear feedback control, a sufficient condition is also presented in the paper. A numerical example is presented at the end of the paper

Quadratic Programming Approach to Fit Protein Complexes into Electron Density Maps

International audienceThe paper investigates the problem of fitting protein complexes into electron density maps. They are represented by high-resolution cryoEM density maps converted into overlapping matrices and partly show a structure of a complex. The general purpose is to define positions of all proteins inside it. This problem is known to be NP-hard, since it lays in the field of combinatorial optimization over a set of discrete states of the complex. We introduce quadratic programming approaches to the problem. To find an approximate solution, we convert a density map into an overlapping matrix, which is generally indefinite. Since the matrix is indefinite, the optimization problem for the corresponding quadratic form is non-convex. To treat non-convexity of the optimization problem, we use different convex relaxations to find which set of proteins minimizes the quadratic form best

Indefinite least squares with a quadratic constraint

An abstract indefinite least squares problem with a quadratic constraint is considered. This is a quadratic programming problem with one quadratic equality constraint, where neither the objective nor the constraint is a convex function. Necessary and sufficient conditions are found for the existence of solutions.Fil: Gonzalez Zerbo, Santiago. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Maestripieri, Alejandra Laura. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería; ArgentinaFil: Martinez Peria, Francisco Dardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería; Argentin

Computational complexity of μ calculation

The structured singular value μ measures the robustness of uncertain systems. Numerous researchers over the last decade have worked on developing efficient methods for computing μ. This paper considers the complexity of calculating μ with general mixed real/complex uncertainty in the framework of combinatorial complexity theory. In particular, it is proved that the μ recognition problem with either pure real or mixed real/complex uncertainty is NP-hard. This strongly suggests that it is futile to pursue exact methods for calculating μ of general systems with pure real or mixed uncertainty for other than small problems

Dealing with degenerated cases in quadratic programming

A new algorithm is described for quadratic programming which is based on a Cholesky factorization that uses a diagonal pivoting strategy and that allows to compute null or negative curvature directions. The algorithm is numerically stable and has shown efficiency solving positive-definite and indefinite problems. It is specially interesting in indefinite cases because the initial point does not need to be a vertex of the feasible set. So we avoid introducing artificial constraints in the problem, which turns out to be very efficient in parametric programming. At the same time techniques for updating matrix factorizations are used