Analytic efficient solution set for multi-criteria quadratic programs

We present active set methods to evaluate the exact analytic efficient solution set for multi-criteria convex quadratic programming problems (MCQP) subject to linear constraints. The idea is based on the observations that a strictly convex programming problem admits a unique solution, and that the efficient solution set for a multi-criteria strictly convex quadratic programming problem with linear equality constraints can be parameterized. The case of bi-criteria quadratic programming (BCQP) is first discussed since many of the underlying ideas can be explained much more clearly in the case of two objectives. In particular we note that the efficient solution set of a BCQP problem is a curve on the surface of a polytope. The extension to problems with more than two objectives is straightforward albeit some slightly more complicated notation. Two numerical examples are given to illustrate the proposed methods.     

A Semidefinite Programming Heuristic for Quadratic Programming Problems with Complementarity Constraints

The presence of complementarity constraints brings a combinatorial flavour to an optimization problem. A quadratic programming problem with complementarity constraints can be relaxed to give a semidefinite programming problem. The solution to this relaxation can be used to generate feasible solutions to the complementarity constraints. A quadratic programming problem is solved for each of these feasible solutions and the best resulting solution provides an estimate for the optimal solution to the quadratic program with complementarity constraints. Computational testing of such an approach is described for a problem arising in portfolio optimization.Research supported in part by the National Science Foundations VIGRE Program (Grant DMS-9983646).Research partially supported by NSF Grant number CCR-9901822.     

最大割问题的二次规划方法

本文给出了最大割问题的二次规划算法。这种算法通过求解最大割问题的二次规划松弛给出了一种较好的界，然后用分支定界法得到了最大割问题的解。数值结果表明这种算法是非常有效的。     

A Dynamical Systems Analysis of Semidefinite Programming with Application to Quadratic Optimization with Pure Quadratic Equality Constraints

This paper considers the problem of minimizing a quadratic cost subject to purely quadratic equality constraints. This problem is tackled by first relating it to a standard semidefinite programming problem. The approach taken leads to a dynamical systems analysis of semidefinite programming and the formulation of a gradient descent flow which can be used to solve semidefinite programming problems. Though the reformulation of the initial problem as a semidefinite pro- gramming problem does not in general lead directly to a solution of the original problem, the initial problem is solved by using a modified flow incorporating a penalty function. Accepted 10 March 1998     

A neural network for solving a convex quadratic bilevel programming problem

A neural network is proposed for solving a convex quadratic bilevel programming problem. Based on Lyapunov and LaSalle theories, we prove strictly an important theoretical result that, for an arbitrary initial point, the trajectory of the proposed network does converge to the equilibrium, which corresponds to the optimal solution of a convex quadratic bilevel programming problem. Numerical simulation results show that the proposed neural network is feasible and efficient for a convex quadratic bilevel programming problem.     

A new descent algorithm for solving quadratic bilevel programming problems

1. IntroductionA bilevel programming problem (BLPP) involves two sequential optimization problems where the constraint region of the upper one is implicitly determined by the solutionof the lower. It is proved in [1] that even to find an approximate solution of a linearBLPP is strongly NP-hard. A number of algorithms have been proposed to solve BLPPs.Among them, the descent algorithms constitute an important class of algorithms for nonlinear BLPPs. However, it is assumed for almost all…     

Analysis of unbonded contact problems by means of quadratic programming

Two-body, elastic, unbonded contact problems are formulated as quadratic programming problems. Uniqueness theorems of quadratic programming theory are applied to show that the solution of a contact problem, if one exists, is unique and can be readily found by the modified simplex method of quadratic programming. A solution technique that is compatible with finite-element methods is developed, so that contact problems with complex boundary configurations can be routinely solved. A number of classical and nonclassical problems are solved. Good agreement is found for problems with previously known solutions.     

Quadratic programming problems and related linear complementarity problems

This paper investigates the general quadratic programming problem, i.e., the problem of finding the minimum of a quadratic function subject to linear constraints. In the case where, over the set of feasible points, the objective function is bounded from below, this problem can be solved by the minimization of a linear function, subject to the solution set of a linear complementarity problem, representing the Kuhn-Tucker conditions of the quadratic problem.To detect in the quadratic problem the unboundedness from below of the objective function, necessary and sufficient conditions are derived. It is shown that, when these conditions are applied, the general quadratic programming problem becomes equivalent to the investigation of an appropriately formulated linear complementarity problem.This research was supported by the Hungarian Research Foundation, Grant No. OTKA/1044.     

On fuzzy random multiobjective quadratic programming

In this paper, a multiobjective quadratic programming problem having fuzzy random coefficients matrix in the objective and constraints and the decision vector are fuzzy pseudorandom variables is considered. First, we show that the efficient solutions of fuzzy quadratic multiobjective programming problems are resolved into series-optimal-solutions of relative scalar fuzzy quadratic programming. Some theorems are proved to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients, having decision vectors as fuzzy variables. At the end, numerical examples are illustrated in the support of the obtained results.     

A scalable FETI-DP algorithm with non-penetration mortar conditions on contact interface

By combining FETI algorithms of dual-primal type with recent results for bound constrained quadratic programming problems, we develop an optimal algorithm for the numerical solution of coercive variational inequalities. The model problem is discretized using non-penetration conditions of mortar type across the potential contact interface, and a FETI-DP algorithm is formulated. The resulting quadratic programming problem with bound constraints is solved by a scalable algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem. Numerical experiments for non-matching meshes across the contact interface confirm the theoretical scalability of the algorithm.     

解带有二次约束二次规划的一个整体优化方法

在本文中，我们提出了一种解带有二次约束二次规划问题（QP）的新算法，这种方法是基于单纯形分枝定界技术，其中包括极小极大问题和线性规划问题作为子问题，利用拉格朗日松弛和投影次梯度方法来确定问题（QP）最优值的下界，在问题（QP）的可行域是n维的条件下，如果这个算法有限步后终止，得到的点必是问题（QP）的整体最优解；否则，该算法产生的点的序列{v^k}的每一个聚点也必是问题（QP）的整体最优解。     

A NEW INEXACT SEQUENTIAL QUADRATIC PROGRAMMING ALGORITHM

This paper represents an inexact sequential quadratic programming (SQP) algorithm which can solve nonlinear programming (NLP) problems. An inexact solution of the quadratic programming subproblem is determined by a projection and contraction method such that only matrix-vector product is required. Some truncated criteria are chosen such that the algorithm is suitable to large scale NLP problem. The global convergence of the algorithm is proved.     

Solving quadratic convex bilevel programming problems using a smoothing method

In this paper, we present a smoothing sequential quadratic programming to compute a solution of a quadratic convex bilevel programming problem. We use the Karush-Kuhn-Tucker optimality conditions of the lower level problem to obtain a nonsmooth optimization problem known to be a mathematical program with equilibrium constraints; the complementary conditions of the lower level problem are then appended to the upper level objective function with a classical penalty. These complementarity conditions are not relaxed from the constraints and they are reformulated as a system of smooth equations by mean of semismooth equations using Fisher-Burmeister functional. Then, using a quadratic sequential programming method, we solve a series of smooth, regular problems that progressively approximate the nonsmooth problem. Some preliminary computational results are reported, showing that our approach is efficient.     

凸二次交叉规划的等价形式

利用参数规划逆问题考虑凸二次交叉规划与多目标规划的关系 ,把交叉规划转变为同变量规划组 ,再把同变量规划组变为多目标规划 ,证明了凸二次交叉规划的均衡解与多目标规划的最优解的关系。     

带非凸二次约束的二次比式和问题的全局优化算法(英文)

对带非凸二次约束的二次比式和问题(P)给出分枝定界算法,首先将问题(P)转化为其等价问题(Q),然后利用线性化技术,建立了(Q)松弛线性规划问题(RLP),通过对(RLP)可行域的细分及求解一系列线性规划问题,不断更新(Q)的上下界,从理论上证明了算法的收敛性,数值实验表明了算法的可行性和有效性.     

One way to solve the parametric quadratic programming problem

A method is presented for the solution of the parametric quadratic programming problem by the use of conjugate directions. It is based on the method for quadratic programming proposed by the author in [1].While engaged in this research the author had a part-time post with the Manpower Services Commission.     

On active-set methods for the quadratic programming problem

The active-set Newton method developed earlier by the authors for mixed complementarity problems is applied to solving the quadratic programming problem with a positive definite matrix of the objective function. A theoretical justification is given to the fact that the method is guaranteed to find the exact solution in a finite number of steps. Numerical results indicate that this approach is competitive with other available methods for quadratic programming problems.     

最优控制问题参数化研究—基于勒让德正交多项式逼近

使用勒让德正交多项式逼近方法,将Lagrange型最优控制问题转化为非线性规划问题.采用序列二次规划方法对此非线性规划问题的求解,并对多项式逼近和非线性规划求解后得到的解是否收敛给出了证明和实例分析.     

A new approach to the weighted peak-constrained least-square error FIR digital filter optimal design problem

This paper deals with the design of linear-phase finite impulse response (FIR) digital filters using weighted peak-constrained least-squares (PCLS) optimization. The PCLS error design problem is formulated as a quadratically constrained quadratic semi-infinite programming problem. An exchange algorithm with a new exchange rule is proposed to solve the problem. The algorithm provides the approximate optimal solution after a finite number of iterations. In particular, the subproblem solved at each iteration is a quadratically constrained quadratic programming. We can rewrite it as a conic optimization problem solvable in polynomial time. For illustration, numerical examples are solved using the proposed algorithm.     

Reducing quadratic programming problem to regression problem: Stepwise algorithm

Quadratic programming is concerned with minimizing a convex quadratic function subject to linear inequality constraints. The variables are assumed to be nonnegative. The unique solution of quadratic programming (QP) problem (QPP) exists provided that a feasible region is non-empty (the QP has a feasible space).A method for searching for the solution to a QP is provided on the basis of statistical theory. It is shown that QPP can be reduced to an appropriately formulated least squares (LS) problem (LSP) with equality constraints and nonnegative variables. This approach allows us to obtain a simple algorithm to solve QPP. The applicability of the suggested method is illustrated with numerical examples.