On mixed-μ synthesis

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An Interior Point Method for Linear Programming, with an Active Set Flavor

It is now well established that, especially on large linear programming problems, the simplex method typically takes up a number of iterations considerably larger than recent interior-points methods in order to reach a solution. On the other hand, at each iteration, the size of the linear system of equations solved by the former can be significantly less than that of the linear system solved by the latter. The algorithm proposed in this paper can be thought of as a compromise between the two extremes: conceptually an interior-point method, it ignores, at each iteration, all constraints except those in a small "active set" (in the dual framework). For sake of simplicity, in this first attempt, an affine scaling algorithm is used and strong assumptions are made on the problem. Global and local quadratic convergence is proved. 1 Introduction and Algorithm Statement Consider the problem the linear programming problem (in dual form) (P ) minimize hc; xi s.t. Ax b; x 2 IR n ; with A a..

Tits, A Simple primal-dual feasible interior-point method for nonlinear programming with monotone descent

We propose and analyze a primal-dual interior point method of the “feasible ” type, with the additional property that the objective function decreases at each iteration. A distinctive feature of the method is the use of different barrier parameter values for each constraint, with the purpose of better steering the constructed sequence away from non-KKT stationary points. Assets of the proposed scheme include relative simplicity of the algorithm and of the convergence analysis, strong global and local convergence properties, and good performance in preliminary tests. In addition, the initial point is allowed to lie on the boundary of the feasible set

An SQP Algorithm For Finely Discretized Continuous Minimax Problems And Other Minimax Problems With Many Objective Functions

. A common strategy for achieving global convergence in the solution of semi-infinite programming (SIP) problems, and in particular of continuous minimax problems, is to (approximately) solve a sequence of discretized problems, with a progressively finer discretization meshes. Finely discretized minimax and SIP problems, as well as other problems with many more objectives /constraints than variables, call for algorithms in which successive search directions are computed based on a small but significant subset of the objectives/constraints, with ensuing reduced computing cost per iteration and decreased risk of numerical difficulties. In this paper, an SQP-type algorithm is proposed that incorporates this idea in the particular case of minimax problems. The general case will be considered in a separate paper. The quadratic programming subproblem that yields the search direction involves only a small subset of the objective functions. This subset is updated at each iteration in such a wa..