BOXCQP: an algorithm for bound constrained convex quadratic problems (original) (raw)

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.

Quadratic Programming with Quadratic Constraints

We give a quick and dirty, but reasonably safe, algorithm for the minimization of a convex quadratic function under convex quadratic constraints. The algorithm minimizes the Lagrangian dual by using a safeguarded Newton method with non-negativity constraints.

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.

A new algorithm for quadratic programming

European Journal of Operational Research, 1987

... Active-set methods also does not lem (we have suboptimal solutions and an active require the solution of large linear programming ... active constraint set by solving a (complementary solutions) and special methods feasibility subproblem) so that the prima1 objec-(Bland s rule ...

QPSchur: A dual, active-set, Schur-complement method for large-scale and structured convex quadratic programming

Optimization and Engineering, 2006

We describe an active-set, dual-feasible Schur-complement method for quadratic programming (QP) with positive definite Hessians. The formulation of the QP being solved is general and flexible, and is appropriate for many different application areas. Moreover, the specialized structure of the QP is abstracted away behind a fixed KKT matrix called K o and other problem matrices, which naturally leads to an object-oriented software implementation. Updates to the working set of active inequality constraints are facilitated using a dense Schur complement, which we expect to remain small. Here, the dual Schur complement method requires the projected Hessian to be positive definite for every working set considered by the algorithm. Therefore, this method is not appropriate for all QPs. While the Schur complement approach to linear algebra is very flexible with respect to allowing exploitation of problem structure, it is not as numerically stable as approaches using a QR factorization. However, we show that the use of fixed-precision iterative refinement helps to dramatically improve the numerical stability of this Schur complement algorithm. The use of the objectoriented QP solver implementation is demonstrated on two different application areas with specializations in each area; large-scale model predictive control (MPC) and reduced-space successive quadratic programming (with several different representations for the reduced Hessian). These results demonstrate that the QP solver can exploit application-specific structure in a computationally efficient and fairly robust manner as compared to other QP solver implementations.

Minimizing Nonconvex Quadratic Functions Subject to Bound Constraints

We present an active-set algorithm for finding a local minimizer to a nonconvex bound-constrained quadratic problem. Our algorithm extends the ideas developed by Dostal and Schoberl that is based on the linear conjugate gradient algorithm for (approximately) solving a linear system with a positive-defi nite coefficient matrix. This is achieved by making two key changes. First, we perform a line search along negative curvature directions when they are encountered in the linear conjugate gradient iteration. Second, we use Lanczos iterations to compute approximations to leftmost eigen-pairs, which is needed to promote convergence to points satisfying certain second-order optimality conditions. Preliminary numerical results show that our method is efficient and robust on nonconvex bound-constrained quadratic problems.

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.

An interior-exterior approach for convex quadratic programming

Applied Numerical Mathematics, 2011

In the last decade, a new class of interior–exterior algorithms for linear programming was developed. The method was based on the use of mixed penalty function with two separate parameters to solve a set of sub-penalized problems associated to the initial problem.To study the necessary optimality conditions, one introduced a new concept of the so-called pseudo-gap to describe fully the optimal primal and dual solutions. Only one Newton iteration is sufficient to approximate the solution of penalized problem which satisfies a criterion of proximity.The purpose of this work is to extend the approach to the convex quadratic programming problems.

On the minimum-norm solution of convex quadratic programming

2021

We discuss some basic concepts and present a numerical procedure for finding the minimum-norm solution of convex quadratic programs (QPs) subject to linear equality and inequality constraints. Our approach is based on a theorem of alternatives and on a convenient characterization of the solution set of convex QPs. We show that this problem can be reduced to a simple constrained minimization problem with a once-differentiable convex objective function. We use finite termination of an appropriate Newton’s method to solve this problem. Numerical results show that the proposed method is efficient.

Augmented Lagrangian applied to convex quadratic problems

Applied Mathematics and Computation, 2008

An algorithm based on the Augmented Lagrangian method is proposed to solve convex quadratic programming problem. The quadratic penalty is considered here. Hence, the Augmented Lagrangian function is quadratic when applied to quadratic programming problem. For this penalty, we show that if the Lagrangian function associated with the original problem is strict convex (or convex), then the hessian matrix of Augmented Lagrangian function is positive definite (or positive semi-definite). Numerical experiments are presented illustrating the performance of the algorithm for the CUTE test set.