Large-scale Quadratic Programming Problems Research Articles

Fix and bound: an efficient approach for solving large-scale quadratic programming problems with box constraints

Fix and bound: an efficient approach for solving large-scale quadratic programming problems with box constraints

An accelerated modulus-based matrix splitting iteration method for mixed-size cell circuits legalization

Mixed-size cell circuits dominate in advanced technology node designs, with attendant increases in layout complexity. The introduction of multi-row-height cells requires additional constraints, such as power line alignment, to be considered in the legalization stage, in addition to eliminating overlap while maintaining the results of the global placement stage as much as possible. In this paper, a three-stage legalization methodology is presented. The first is cell preprocessing, namely cell diffusion based on the network flow algorithm. Then the legalization problem is modeled as a quadratic programming problem, which is converted into a linear complementarity problem and solved by an accelerated modulus-based matrix splitting iteration method. Finally, the illegal cells are reshaped. Experimental results reveal that our method achieves 11.8% and 0.4% reduction in total displacement, respectively, over existing two state-of-the-art works, and is 1.407x faster than the modulus-based matrix splitting iteration method. In particular, the proposed approach provides a reference for solving large-scale quadratic programming problems.

Dual Forecasting Algorithm for Technological Structural Matrices in Dynamic Input-Output Models

Based on the global Krotov successive improvement method, we propose a dual computational algorithm for a discrete optimal control problem corresponding to a convex large-scale quadratic programming problem with a separable functional that arises in the prediction of the direct costs (structural) matrix in dynamic input-output models. With decomposition, we are able to use a special form of the constraint matrix to reduce the problem dimension.

A conjugate gradient-based algorithm for large-scale quadratic programming problem with one quadratic constraint

In this paper, we consider the nonconvex quadratically constrained quadratic programming (QCQP) with one quadratic constraint. By employing the conjugate gradient method, an efficient algorithm is proposed to solve QCQP that exploits the sparsity of the involved matrices and solves the problem via solving a sequence of positive definite system of linear equations after identifying suitable generalized eigenvalues. Specifically, we analyze how to recognize hard case (case 2) in a preprocessing step, fixing an error in Sect. 2.2.2 of Pong and Wolkowicz (Comput Optim Appl 58(2):273–322, 2014) which studies the same problem with the two-sided constraint. Some numerical experiments are given to show the effectiveness of the proposed method and to compare it with some recent algorithms in the literature.

On the Solution of Bi-level Large Scale Quadratic Programming Problem with Stochastic Parameters in the Constraints

In this paper, the bi-level large scale quadratic programming problem with stochastic parameters in the constraints (SBLLSQPP) is solved. Randomness is assumed only on the right-hand side of the constraints and the random variables are assumed to be normally distributed. The main features of the proposed solution procedure are based on convert the probabilistic nature of this problem into an equivalent deterministic, Taylor transformation and decomposition algorithms. An illustrative numerical example is given to clarify the main results developed in the paper.

Interior-Point Solver for Large-Scale Quadratic Programming Problems with Bound Constraints

In this paper, we present an interior-point algorithm for large and sparse convex quadratic programming problems with bound constraints. The algorithm is based on the potential reduction method and the use of iterative techniques to solve the linear system arising at each iteration. The global convergence properties of the potential reduction method are reassessed in order to take into account the inexact solution of the inner system. We describe the iterative solver, based on the conjugate gradient method with a limited-memory incomplete Cholesky factorization as preconditioner. Furthermore, we discuss some adaptive strategies for the fill-in and accuracy requirements that we use in solving the linear systems in order to avoid unnecessary inner iterations when the iterates are far from the solution. Finally, we present the results of numerical experiments carried out to verify the effectiveness of the proposed strategies. We consider randomly generated sparse problems without a special structure. Also, we compare the proposed algorithm with the MOSEK solver.

Protein Image Alignment via Piecewise Affine Transformations

We present a new approach for aligning families of 2D gels. Instead of choosing one of the gels as reference and performing a pairwise alignment, we construct an ideal gel that is representative of the entire family and obtain a set of piecewise affine transformations that optimally align each gel of the family to the ideal gel. The coefficients defining the transformations as well as the ideal landmarks are obtained as the solution of a large-scale quadratic programming problem that can be solved efficiently by interior-point methods.

A decomposition algorithm for solving large-scale quadratic programming problems

In this paper an algorithm for solving large-scale programming is proposed. We decompose a large-scale quadratic programming into a serial of small-scale ones and then approximate the solution of the large-scale quadratic programming via the solutions of these small-scale ones. It is proved that the accumulation point of the iterates generated by the algorithm is a global minimum point of the quadratic programming. The algorithm has performed very well in numerical testing. It is a new way of solving large-scale quadratic programming problems.

An iterative working-set method for large-scale nonconvex quadratic programming

We consider a working-set method for solving large-scale quadratic programming problems for which there is no requirement that the objective function be convex. The methods are iterative at two levels, one level relating to the selection of the current working set, and the second due to the method used to solve the equality-constrained problem for this working set. A preconditioned conjugate gradient method is used for this inner iteration, with the preconditioner chosen especially to ensure feasibility of the iterates. The preconditioner is updated at the conclusion of each outer iteration to ensure that this feasibility requirement persists. The well-known equivalence between the conjugate-gradient and Lanczos methods is exploited when finding directions of negative curvature. Details of an implementation—the Fortran 90 package QPA in the forthcoming GALAHAD library—are given.

Structured neural networks for constrained model predictive control

High computational demand in solving the optimization problems associated with the model predictive control (MPC) schemes is a major obstacle when applying the methods to large-scale or fast-sampling systems. In this paper, we propose a new structured neural network approach to solving the quadratic programming problem in the constrained MPC. This new approach has the advantage of solving large-scale quadratic programming problems in a massively parallel fashion. The structured neural network consists of a projection network and a network for implementing the gradient projection algorithm, where the projection network is constructed from specially structured linear neurons with a special training algorithm. We prove that the training algorithm converges to the optimal solution. Finally, we test the method on a simplified paper machine model.

Application of the modified barrier method in large-scale quadratic programming problems

The application of the penalty/modified barrier function method (PE/MBF) is presented for the solution of large-scale positive-semidefinite quadratic programming problems (QP). A review of the recent literature on QP methods is presented and the choice of the PE/MBF method for QP problems is justified by previous experience in very large-scale bound-constrained problems. The proposed algorithm performs two types of iterations: an outer iteration in which the Lagrange multipliers of the bounds are adjusted, and an inner iteration for the solution of an equality constrained subproblem. The inner iteration solves a modified problem, containing penalty/modified barrier terms for the bounds in the objective, and is subject to equality constraints only. The equality constraints are handled directly via the use of additional Lagrange multipliers during the inner iteration and thus, instead of an unconstrained problem, the inner iteration solves a modified equality constrained problem. Any inequality constraints, other than bounds, are formulated as equalities via the use of slack variables. Computational results show this method to be promising, and motivate further investigation for the general case of nonlinear programming problems.

An interior point method for general large-scale quadratic programming problems

In this paper, we present an interior point algorithm for solving both convex and nonconvex quadratic programs. The method, which is an extension of our interior point work on linear programming problems efficiently solves a wide class of largescale problems and forms the basis for a sequential quadratic programming (SQP) solver for general large scale nonlinear programs. The key to the algorithm is a three-dimensional cost improvement subproblem, which is solved at every interation. We have developed an approximate recentering procedure and a novel, adaptive big-M Phase I procedure that are essential to the sucess of the algorithm. We describe the basic method along with the recentering and big-M Phase I procedures. Details of the implementation and computational results are also presented.

Application of the modified barrier method in large-scale Quadratic Programming problems

In this paper we present the application of the Penalty/Modified Barrier Function method (PE/MBF) in the solution of large-scale Quadratic Programming problems (QP). A review of the recent literature on PE/MBF methods is presented and the choice of this method for QP problems is justified by previous experience in very large-scale bound-constrained problems. The proposed algorithm performs two types of iterations: an outer iteration in which the Lagrange multipliers of the bounds are adjusted, and an inner iteration for the solution of an equality constrained subproblem. The inner iteration solves a modified problem, containing penalty/modified barrier terms for the bounds in the objective and is subject to equality constraints only. The equality constraints are handled directly via the use of additional Lagrange multipliers during the inner iteration and thus, instead of an unconstrained problem, the inner iteration solves a modified equality constrained problem. Any inequality constraints, other than bounds, are formulated as equalities via the use of slack variables. Computational results show this method to be promising, and motivate further investigation for the general case of nonlinear programming problems.

A BLOCK-PARALLEL CONJUGATE GRADIENT METHOD FOR SEPARABLE QUADRATIC PROGRAMMING PROBLEMS^1

For a large-scale quadratic programming problem with separable objective function, a variant of the conjugate gradient method can effectively be applied to the dual problem. In this paper, we consider a block-parallel modification of the conjugate gradient method, which is suitable for implementation on a parallel computer. More precisely, the method proceeds in a block Jacobi manner and executes the conjugate gradient iteration to solve quadratic programming subproblems associated with respective blocks. We implement the method on a Connection Machine Model CM-5 in the Single-Program Multiple-Data model of computation. We report some numerical results, which show that the proposed method is effective particularly for problems with some block structure.

Solution of Spatial Equilibrium Problems with Benders Decomposition

Spatial equilibrium problems are frequently formulated as large scale quadratic programming problems or linear complementarity problems. We show that these problems can be reduced to two or more smaller problems with Generalized Benders Decomposition. The procedure then becomes iterative with the repetitive solution of the smaller problems. In practice, the iterative procedure has converged rapidly.