A new algorithm for quadratic programming (original) (raw)

A feasible active set method for strictly convex quadratic problems with simple bounds *

SIAM Journal on Optimization

A primal-dual active set method for quadratic problems with bound constraints is presented which extends the infeasible active set approach of [K. Kunisch and F. Rendl. An infeasible active set method for convex problems with simple bounds. SIAM Journal on Optimization, 14(1):35-52, 2003]. Based on a guess of the active set, a primal-dual pair (x,α) is computed that satisfies stationarity and the complementary condition. If x is not feasible, the variables connected to the infeasibilities are added to the active set and a new primal-dual pair (x,α) is computed. This process is iterated until a primal feasible solution is generated. Then a new active set is determined based on the feasibility information of the dual variable α. Strict convexity of the quadratic problem is sufficient for the algorithm to stop after a finite number of steps with an optimal solution. Computational experience indicates that this approach also performs well in practice.

qpOASES: a parametric active-set algorithm for quadratic programming

Mathematical Programming Computation, 2014

Many practical applications lead to optimization problems that can either be stated as quadratic programming (QP) problems or require the solution of QP problems on a lower algorithmic level. One relatively recent approach to solve QP problems are parametric active-set methods that are based on tracing the solution along a linear homotopy between a QP problem with known solution and the QP problem to be solved. This approach seems to make them particularly suited for applications where a-priori information can be used to speed-up the QP solution or where high solution accuracy is required. In this paper we describe the open-source C++ software package qpOASES, which implements a parametric active-set method in a reliable and efficient way. Numerical tests show that qpOASES can outperform other popular academic and commercial QP solvers on small-to medium-scale convex test examples of the Maros-Mészáros QP collection. Moreover, various interfaces to third-party software packages make it easy to use, even on embedded computer hardware. Finally, we describe how qpOASES can be used to compute critical points of nonconvex QP problems.

A Feasible Active Set Method with Reoptimization for Convex Quadratic Mixed-Integer Programming

SIAM Journal on Optimization, 2016

We propose a feasible active set method for convex quadratic programming problems with non-negativity constraints. This method is specifically designed to be embedded into a branch-and-bound algorithm for convex quadratic mixed integer programming problems. The branch-and-bound algorithm generalizes the approach for unconstrained convex quadratic integer programming proposed by Buchheim, Caprara and Lodi [8] to the presence of linear constraints. The main feature of the latter approach consists in a sophisticated preprocessing phase, leading to a fast enumeration of the branch-and-bound nodes. Moreover, the feasible active set method takes advantage of this preprocessing phase and is well suited for reoptimization. Experimental results for randomly generated instances show that the new approach significantly outperforms the MIQP solver of CPLEX 12.6 for instances with a small number of constraints.

Active Set Methods with Reoptimization for Convex Quadratic Integer Programming

Springer eBooks, 2014

We present a fast branch-and-bound algorithm for solving convex quadratic integer programs with few linear constraints. In each node, we solve the dual problem of the continuous relaxation using an infeasible active set method proposed by Kunisch and Rendl [11] to get a lower bound; this active set algorithm is well suited for reoptimization. Our algorithm generalizes a branch-and-bound approach for unconstrained convex quadratic integer programming proposed by Buchheim, Caprara and Lodi [5] to the presence of linear constraints. The main feature of the latter approach consists in a sophisticated preprocessing phase, leading to a fast enumeration of the branch-and-bound nodes. Experimental results for randomly generated instances are presented. The new approach significantly outperforms the MIQP solver of CPLEX 12.4 for instances with a small number of constraints.

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.

A Dynamic Active-Set Method for Linear Programming

American Journal of Operations Research, 2015

An efficient active-set approach is presented for both nonnegative and general linear programming by adding varying numbers of constraints at each iteration. Computational experiments demonstrate that the proposed approach is significantly faster than previous active-set and standard linear programming algorithms.

An active-set strategy in an interior point method for linear programming

Mathematical Programming, 1993

We will present a potential reduction method for linear programming where only the constraints with relatively small dual slacks--termed "active constraints"--will be taken into account to form the ellipsoid constraint at each iteration of the process. The algorithm converges to the optimal feasible solution in O(,/~L) iterations with the same polynomial bound as in the full constraints case, where n is the number of variables and L is the data length. If a small portion of the constraints is active near the optimal solution, the computational cost to find the next direction of movement in one iteration may be considerably reduced by the proposed strategy.

An O(n2) active set algorithm for solving two related box constrained parametric quadratic programs

Numerical Algorithms, 2001

Recently, O(n 2) active set methods have been presented for minimizing the parametric quadratic functions (1/2)x Dx − a x + λ|γ x − c| and (1/2)x Dx − a x + (λ/2)(γ x − c) 2 , respectively, subject to l x b, for all nonnegative values of the parameter λ. Here, D is a positive diagonal n × n matrix, γ and a are arbitrary n-vectors, c is an arbitrary scalar; l and b are arbitrary n-vectors such that l b. In this paper, we show that each one of these algorithms may be used to simultaneously solve both parametric programs with no additional computational cost.

AnO(n 2) active set method for solving a certain parametric quadratic program

Journal of Optimization Theory and Applications, 1992

This paper presents an O(n 2) method for solving the parametric quadratic program rain(1/2)x'Dx-a'x + (M2) 7 j x j-c , J having lower and upper bounds on the variables, for all nonnegative values of the parameter ~. Here, D is a positive diagonal matrix, a an arbitrary n-vector, each 7J, J= 1. .. .. n, and c are arbitrary scalars. An application to economics is also presented.