An interior-exterior approach for convex quadratic programming (original) (raw)

A primal–dual regularized interior-point method for convex quadratic programs

Mathematical Programming Computation, 2012

Interior-point methods in augmented form for linear and convex quadratic programming require the solution of a sequence of symmetric indefinite linear systems which are used to derive search directions. Safeguards are typically required in order to handle free variables or rank-deficient Jacobians. We propose a consistent framework and accompanying theoretical justification for regularizing these linear systems. Our approach can be interpreted as a simultaneous proximal-point regularization of the primal and dual problems. The regularization is termed exact to emphasize that, although the problems are regularized, the algorithm recovers a solution of the original problem, for appropriate values of the regularization parameters.

A primal‐dual interior-point method for linear optimization based on a new proximity function

Optimization Methods and Software, 2002

In this paper we present a generic primal-dual interior-point algorithm for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. We present some powerful tools for the analysis of the algorithm under the assumption that the kernel function satisfies three easy to check and mild conditions (i.e., exponential convexity, superconvexity and monotonicity of the second derivative). The approach is demonstrated by introducing a new kernel function and showing that the corresponding large-update algorithm improves the iteration complexity with a factor n 1 4 when compared with the classical method, which is based on the use of the logarithmic barrier function.

An exterior Newton method for strictly convex quadratic programming

2000

We propose an exterior Newton method for strictly convex quadratic programming (QP) problems. This method is based on a dual formulation: a sequence of points is generated which monotonically decreases the dual objective function. We show that the generated sequence converges globally and quadratically to the solution (if the QP is feasible and certain nondegeneracy assumptions are satisfied). Measures for detecting infeasibility are provided. The major computation in each iteration is to solve a KKT-like system. Therefore, given an effective symmetric sparse linear solver, the proposed method is suitable for large sparse problems. Preliminary numerical results are reported.

Convergence Analysis of an Inexact Interior-Point Quadratic Programming Algorithm

2004

In this paper we develop an interior-point primal-dual long-step path-following algorithm for convex quadratic programming (CQP) whose search directions are computed by means of an iterative (linear system) solver. We propose a new linear system, which we refer to as the augmented normal equation (ANE), to determine the primal-dual search directions. Since the condition number of the matrix associated with the ANE may become large for degenerate CQP problems, we use a maximum weight basis preconditioner introduced in [16, 14] to better condition this matrix. Using a result obtained in [13], we establish a uniform bound, depending only on CQP data, on the number of iterations required for the iterative solver to obtain a suciently accurate solution to the ANE. Since the iterative solver can only generate an approximate solution to the ANE, this solution does not yield a primal-dual search direction satisfying all equations of the primal-dual Newton system. We propose a way to compute...

A Primal-Dual Interior Point Algorithm for Linear Programming

Progress in Mathematical Programming, 1989

This chapter presents an algorithm that works simultaneously on primal and dual linear programming problems and generates a sequence of pairs of their interior feasible solutions. Along the sequence generated, the duality gap converges to zero at least linearly with a global convergence ratio (1-Yf/n); each iteration reduces the duality gap by at least Yf/n. Here n denotes the size of the problems and Yf a positive number depending on initial interior feasible solutions of the problems. The algorithm is based on an application of the classical logarithmic barrier function method to primal and dual linear programs, which has recently been proposed and studied by Megiddo. N. Megiddo (ed.

An interior point-proximal method of multipliers for convex quadratic programming

Computational Optimization and Applications

In this paper we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving linearly constrained convex quadratic programming problems. We apply few iterations of the interior point method to each sub-problem of the proximal method of multipliers. Once a satisfactory solution of the PMM sub-problem is found, we update the PMM parameters, form a new IPM neighbourhood and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under standard assumptions. To our knowledge, this is the first polynomial complexity result for a primal-dual regularized IPM. The algorithm is guided by the use of a single penalty parameter; that of the logarithmic barrier. In other words, we show that IP-PMM inherits the polynomial complexity of IPMs, as well as the strict convexity of the PMM sub-problems. The updates of the penalty ...

A Dual Method For Solving General Convex Quadratic Programs

2009

In this paper, we present a new method for solving quadratic programming problems, not strictly convex. Constraints of the problem are linear equalities and inequalities, with bounded variables. The suggested method combines the active-set strategies and support methods. The algorithm of the method and numerical experiments are presented, while comparing our approach with the active set method on randomly generated problems.

Primal—dual methods for linear programming

Mathematical Programming, 1995

Many interior-point methods for linear programming are based on the properties of the logarithmic barrier function. We first give a convergence proof for the (primal) projected Newton barrier method. We then analyze three types of barrier method that can be categorized as primal, dual and primal-dual. All three approaches may be derived from the application of Newton's method to different variants of the same system of nonlinear equations. A fourth variant of the same equations leads to a new primal-dual algorithm.