New infeasible interior-point algorithm based on monomial method

SIAM Journal on Optimization, 2005

Interior point methods for nonlinear programs (NLP) are adapted for solution of mathematical programs with complementarity constraints (MPCCs). The constraints of the MPCC are suitably relaxed so as to guarantee a strictly feasible interior for the inequality constraints. The standard primal-dual algorithm has been adapted with a modified step calculation. The algorithm is shown to be superlinearly convergent in the neighborhood of the solution set under assumptions of MPCC-LICQ, strong stationarity and upper level strict complementarity. The modification can be easily accommodated within most nonlinear programming interior point algorithms with identical local behavior. Numerical experience is also presented and holds promise for the proposed method.

Computational Optimization and Applications, 2012

The paper shows that the global resolution of a general convex quadratic program with complementarity constraints (QPCC), possibly infeasible or unbounded, can be accomplished in finite time. The method constructs a minmax mixed integer formulation by introducing finitely many binary variables, one for each complementarity constraint. Based on the primal-dual relationship of a pair of convex quadratic programs and on a logical Benders scheme, an extreme ray/point generation procedure is developed, which relies on valid satisfiability constraints for the integer program. To improve this scheme, we propose a two-stage approach wherein the first stage solves the mixed integer quadratic program with pre-set upper bounds on the complementarity variables, and the second stage solves the program outside this bounded region by the Benders scheme. We report computational results with our method. We also investigate the addition of a penalty term y T Dw to the objective function, where y and w are the complementary variables and D is a nonnegative diagonal matrix. The matrix D can be chosen effectively by solving a semidefinite program, ensuring that the objective function remains convex. The addition of the penalty term can often reduce the overall runtime by at least 50%. We report preliminary computational testing on a QP relaxation method which can be used to obtain better lower bounds from infeasible points; this method could be incorporated into a branching scheme. By combining the penalty method and the QP relaxation method, more than 90% of the gap can be closed for some QPCC problems.

Lecture Notes in Computer Science, 1991

Croatian Operational Research Review, 2018

In this paper, a feasible primal-dual path-following interior-point algorithm for monotone semidefinite linear complementarity problems is proposed. At each iteration, the algorithm uses only full Nesterov-Todd feasible steps for tracing approximately the central-path and getting an approximated solution of this problem. Under a new appropriate choices of the threshold τ which defines the size of the neighborhood of the central-path and of the update barrier parameter θ, we show that the algorithm is well-defined and enjoys the locally quadratically convergence. Moreover, we prove that the short-step algorithm deserves the best known iteration bound, namely, O(√ n log n). Finally, some numerical results are reported to show the practical performance of the 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...

Journal of New Researches in Mathematics, 2016

In this paper, we propose a feasible interior-point algorithm for mixed symmetric cone linear complementarity problems which are a general class of complementarity problems. The symmetrization of the search directions used in this paper is based on Nesterov and Todd scaling scheme. By using Euclidean Jordan algebra, we prove the convergence analysis of the proposed algorithm and show that the complexity bound of the algorithm matches the currently best known iteration bound for feasible interior-point methods.

This final project offers an alternative to solve convex quadratic programming optimization problem using Interior Point method. This method cuts across the interior of the solution space to optimal solution. Unfortunately, this method can not provide an estimation in total number of iterations produced.

2014

A full Nesterov-Todd (NT) step infeasible interior-point algorithm is proposed for solving monotone linear complementarity problems over symmetric cones by using Euclidean Jordan algebra. Two types of full NT-steps are used, feasibility steps and centering steps. The algorithm starts from strictly feasible iterates of a per- turbed problem, and, using the central path and feasibility steps, nds strictly feasible iterates for the next perturbed problem. By using centering steps for the new perturbed problem, strictly fea- sible iterates are obtained to be close enough to the central path of the new perturbed problem. The starting point depends on two positive numbersp andd. The algorithm terminates either by nding an ϵ-solution or detecting that the symmetric cone linear complementarity problem has no optimal solution with vanishing duality gap satisfying a condition in terms ofp andd. The it- eration bound coincides with the best known bound for infeasible interior-point methods. Ke...

INFORMS Journal on Computing, 2017

We present a homogeneous algorithm equipped with a modified potential function for the monotone complementarity problem. We show that this potential function is reduced by at least a constant amount if a scaled Lipschitz condition (SLC) is satisfied. A practical algorithm based on this potential function is implemented in a software package named iOptimize. The implementation in iOptimize maintains global linear and polynomial time convergence properties, while achieving practical performance. It either successfully solves the problem, or concludes that the SLC is not satisfied. When compared with the mature software package MOSEK (barrier solver version 6.0.0.106), iOptimize solves convex quadratic programming problems, convex quadratically constrained quadratic programming problems, and general convex programming problems in fewer iterations. Moreover, several problems for which MOSEK fails are solved to optimality. We also find that iOptimize detects infeasibility more reliably t...