An interior point method for quadratic programs based on conjugate projected gradients

We propose an interior point method for large-scale convex quadratic programming where no assumptions are made about the sparsity structure of the quadratic coefficient matrixQ. The interior point method we describe is a doubly iterative algorithm that invokes aconjugate projected gradient procedure to obtain the search direction. The effect is thatQ appears in a conjugate direction routine rather than in a matrix factorization. By doing this, the matrices to be factored have the same nonzero structure as those in linear programming. Further, one variant of this method istheoretically convergent with onlyone matrix factorization throughout the procedure.     

A big-M type method for the computation of projections onto polyhedrons

In a previous work (Ref. 1), we examined some active set methods for the computation of the projection of a point onto a polyhedron when a feasible point is known. In this paper, we assume that such a point is not known and examine a method similar to the big-M method developed for the solution of linear programming problems. Special attention is given to the study of computing error propagation.This research was supported partially by the Progetto Finalizzato Informatica del CNR, Sottoprogetto P1, Sofmat.     

An SQP algorithm for extended linear-quadratic problems in stochastic programming

Extended Linear-Quadratic Programming (ELQP) problems were introduced by Rockafellar and Wets for various models in stochastic programming and multistage optimization. Several numerical methods with linear convergence rates have been developed for solving fully quadratic ELQP problems, where the primal and dual coefficient matrices are positive definite. We present a two-stage sequential quadratic programming (SQP) method for solving ELQP problems arising in stochastic programming. The first stage algorithm realizes global convergence and the second stage algorithm realizes superlinear local convergence under a condition calledB-regularity.B-regularity is milder than the fully quadratic condition; the primal coefficient matrix need not be positive definite. Numerical tests are given to demonstrate the efficiency of the algorithm. Solution properties of the ELQP problem underB-regularity are also discussed.Supported by the Australian Research Council.     

A strictly improving linear programming Phase I algorithm

Instead of trying to recognize and avoid degenerate steps in the simplex method (as some variants do), we have developed a new Phase I algorithm that is impervious to degeneracy. The new algorithm solves a non-negative least-squares problem in order to find a Phase I solution. In each iteration, a simple two-variable least-squares subproblem is used to select an incoming column to augment a set of independent columns (called basic) to get a strictly better fit to the right-hand side. Although this is analogous in many ways to the simplex method, it can be proved that strict improvement is attained at each iteration, even in the presence of degeneracy. Thus cycling cannot occur, and convergence is guaranteed. This algorithm is closely related to a number of existing algorithms proposed for non-negative least-squares and quadratic programs.When used on the 30 smallest NETLIB linear programming test problems, the computational results for the new Phase I algorithm were almost 3.5 times faster than a particular implementation of the simplex method; on some problems, it was over 10 times faster. Best results were generally seen on the more degenerate problems.     

An Infeasible Mizuno–Todd–Ye Type Algorithm for Convex Quadratic Programming with Polynomial Complexity

Infeasible interior point methods have been very popular and effective. In this paper, we propose a predictor–corrector infeasible interior point algorithm for convex quadratic programming, and we prove its convergence and analyze its complexity. The algorithm has the polynomial numerical complexity with O(nL)-iteration.     

An extension of Karmarkar's projective algorithm for convex quadratic programming

We present an extension of Karmarkar's linear programming algorithm for solving a more general group of optimization problems: convex quadratic programs. This extension is based on the iterated application of the objective augmentation and the projective transformation, followed by optimization over an inscribing ellipsoid centered at the current solution. It creates a sequence of interior feasible points that converge to the optimal feasible solution in O(Ln) iterations; each iteration can be computed in O(Ln ³) arithmetic operations, wheren is the number of variables andL is the number of bits in the input. In this paper, we emphasize its convergence property, practical efficiency, and relation to the ellipsoid method.     

A randomized scheme for speeding up algorithms for linear and convex programming problems with high constraints-to-variables ratio

We extend Clarkson's randomized algorithm for linear programming to a general scheme for solving convex optimization problems. The scheme can be used to speed up existing algorithms on problems which have many more constraints than variables. In particular, we give a randomized algorithm for solving convex quadratic and linear programs, which uses that scheme together with a variant of Karmarkar's interior point method. For problems withn constraints,d variables, and input lengthL, ifn = (d ²), the expected total number of major Karmarkar's iterations is O(d ²(logn)L), compared to the best known deterministic bound of O( L). We also present several other results which follow from the general scheme.     

Symmetric indefinite systems for interior point methods

We present a unified framework for solving linear and convex quadratic programs via interior point methods. At each iteration, this method solves an indefinite system whose matrix is instead of reducing to obtain the usualAD ² A ^T system. This methodology affords two advantages: (1) it avoids the fill created by explicitly forming the productAD ² A ^T whenA has dense columns; and (2) it can easily be used to solve nonseparable quadratic programs since it requires only thatD be symmetric. We also present a procedure for converting nonseparable quadratic programs to separable ones which yields computational savings when the matrix of quadratic coefficients is dense.     

Efficient Sequential Quadratic Programming Implementations for Equality-Constrained Discrete-Time Optimal Control

Efficient sequential quadratic programming (SQP) implementations are presented for equality-constrained, discrete-time, optimal control problems. The algorithm developed calculates the search direction for the equality-based variant of SQP and is applicable to problems with either fixed or free final time. Problem solutions are obtained by solving iteratively a series of constrained quadratic programs. The number of mathematical operations required for each iteration is proportional to the number of discrete times N. This is contrasted by conventional methods in which this number is proportional to N ³. The algorithm results in quadratic convergence of the iterates under the same conditions as those for SQP and simplifies to an existing dynamic programming approach when there are no constraints and the final time is fixed. A simple test problem and two application problems are presented. The application examples include a satellite dynamics problem and a set of brachistochrone problems involving viscous friction.     

A Class of Infeasible Interior Point Algorithms for Convex Quadratic Programming

This article proposes a class of infeasible interior point algorithms for convex quadratic programming, and analyzes its complexity. It is shown that this algorithm has the polynomial complexity. Its best complexity is O(nL).     

A Simplicial Branch-and-Bound Method for Solving Nonconvex All-Quadratic Programs

In this paper we present an algorithm for solving nonconvex quadratically constrained quadratic programs (all-quadratic programs). The method is based on a simplicial branch-and-bound scheme involving mainly linear programming subproblems. Under the assumption that a feasible point of the all-quadratic program is known, the algorithm guarantees an -approximate optimal solution in a finite number of iterations. Computational experiments with an implementation of the procedure are reported on randomly generated test problems. The presented algorithm often outperforms a comparable rectangular branch-and-bound method.     

A Sequential Quadratically Constrained Quadratic Programming Method of Feasible Directions

In this paper, a sequential quadratically constrained quadratic programming method of feasible directions is proposed for the optimization problems with nonlinear inequality constraints. At each iteration of the proposed algorithm, a feasible direction of descent is obtained by solving only one subproblem which consist of a convex quadratic objective function and simple quadratic inequality constraints without the second derivatives of the functions of the discussed problems, and such a subproblem can be formulated as a second-order cone programming which can be solved by interior point methods. To overcome the Maratos effect, an efficient higher-order correction direction is obtained by only one explicit computation formula. The algorithm is proved to be globally convergent and superlinearly convergent under some mild conditions without the strict complementarity. Finally, some preliminary numerical results are reported. Project supported by the National Natural Science Foundation (No. 10261001), Guangxi Science Foundation (Nos. 0236001, 064001), and Guangxi University Key Program for Science and Technology Research (No. 2005ZD02) of China.     

求解凸二次规划问题的势下降内点算法

1　引　言二次规划问题的求解是数学规划和工业应用等领域的一个重要课题 ,同时也是解一般非线性规划问题的序列二次规划算法的关键 .求解二次规划问题的早期技术是利用线性规划问题的单纯形方法求解二次规划问题的 KKT最优性必要条件[1 ] .这类算法比较直观 ,但在处理不等式约束时 ,松弛变量的引进很容易导致求解过程的明显减慢 .有效集策略是求解二次规划问题的另一类主要技术 .这类方法一般都是稳定的 ,但随着问题中大量不等式约束的出现 ,其收敛速度将越来越低[2 ] .简约空间技术将所求问题的 Hessian阵投影到自由变量所在的子空间中 …     

Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound

We consider a recent branch-and-bound algorithm of the authors for nonconvex quadratic programming. The algorithm is characterized by its use of semidefinite relaxations within a finite branching scheme. In this paper, we specialize the algorithm to the box-constrained case and study its implementation, which is shown to be a state-of-the-art method for globally solving box-constrained nonconvex quadratic programs. S. Burer was supported in part by NSF Grants CCR-0203426 and CCF-0545514.     

不等式约束二次规划的一新算法

文献[1]提出了一般等式约束非线性规划问题一种求解途径．文献[2]应用这一途径给出了等式约束二次规划问题的一种算法，本文在文献[1]和[2]的基础上对不等式约束二次规划问题提出了一种新算法．     

An RQP algorithm using a differentiable exact penalty function for inequality constrained problems

In this paper we propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function. The algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an Armijo-type line search procedure that avoids the need to evaluate second order derivatives of the problem functions. We prove that the algorithm possesses global and superlinear convergence properties. Numerical results are reported.     

A trust region SQP algorithm for mixed-integer nonlinear programming

We propose a modified sequential quadratic programming method for solving mixed-integer nonlinear programming problems. Under the assumption that integer variables have a smooth influence on the model functions, i.e., that function values do not change drastically when in- or decrementing an integer value, successive quadratic approximations are applied. The algorithm is stabilized by a trust region method with Yuan’s second order corrections. It is not assumed that the mixed-integer program is relaxable or, in other words, function values are evaluated only at integer points. The Hessian of the Lagrangian function is approximated by a quasi-Newton update formula subject to the continuous and integer variables. Numerical results are presented for a set of 80 mixed-integer test problems taken from the literature. The surprising result is that the number of function evaluations, the most important performance criterion in practice, is less than the number of function calls needed for solving the corresponding relaxed problem without integer variables.     

Interior path following primal-dual algorithms. part II: Convex quadratic programming

We describe a primal-dual interior point algorithm for convex quadratic programming problems which requires a total of number of iterations, whereL is the input size. Each iteration updates a penalty parameter and finds an approximate Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier function problem. The algorithm is based on the path following idea. The total number of arithmetic operations is shown to be of the order of O(n ³ L).     

A sparse sequential quadratic programming algorithm

Described here is the structure and theory for a sequential quadratic programming algorithm for solving sparse nonlinear optimization problems. Also provided are the details of a computer implementation of the algorithm along with test results. The algorithm maintains a sparse approximation to the Cholesky factor of the Hessian of the Lagrangian. The solution to the quadratic program generated at each step is obtained by solving a dual quadratic program using a projected conjugate gradient algorithm. An updating procedure is employed that does not destroy sparsity.