Reference EPFL-ARTICLE-229257 URL: http: arxiv.org abs 0903.4856 Record created on 2017-06-21, modified on 2017-06-21
We present an ”active set” algorithm for the solution of the convex (but not necessarily strictly convex) parametric quadratic programming problem. The optimal solution and associated multipliers are obtained as piece-wise linear functions of the parameter. At the end of each interval, the active set is changed by either adding, deleting, or exchanging a constraint. The method terminates when either the optimal solution has been obtained for all values of the parameter, or, a further increase in the parameter results in either the feasible region being null or the objective function being unbounded from below. The method used to solve the linear equations associated with a particular active set is left unspecified. The parametric algorithm can thus be implemented using the linear equation solving method of any active set quadratic programming algorithm. Abstract : An algorithm is described for determining the optimal solution of parametric linear and quadratic programming problems as an explicit piecewise linear function of the parameter. Each linear function is uniquely determined by an appropriate subset of active constraints. For every critical value of the parameter a new subset has to be determined. A simple rule is given for adding and deleting constraints from this subset. (Author) A method is presented for the solution of the parametric quadratic programming problem by the use of conjugate directions. It is based on the method for quadratic programming proposed by the author in [1]. The main contributions in this thesis are advances in parametric programming. The thesis is divided into three parts; theoretical advances, application areas and constrained control allocation. The first part deals with continuity properties and the structure of solutions to convex parametric quadratic and linear programs. The second part focuses on applications of parametric quadratic and linear programming in control theory. The third part deals with constrained control allocation and how parametric programming can be used to obtain explicit solutions to this problem.  In this paper we consider a semidefinite programming (SDP) problem in which the objective function depends linearly on a scalar parameter. We study the properties of the optimal objective function value as a function of that parameter and extend the concept of the optimal partition and its range in linear programming to SDP. We also consider an approach to sensitivity analysis in SDP and the extension of our results to an SDP problem with a parametric right-hand side.