The Minimum Sum of Squares Change to Univariate Data that gives Convexity

Let n measurements of a real valued function of one variablebe given. If the function is convex but the data have lost convexitydue to the errors of the measuring process, then the least sumof squares change to the data that provides nonnegative seconddivided differences may be required. An algorithm is proposedfor this highly structured quadratic programming calculation.First a procedure that requires only O(n) computer operationsgenerates a starting point for the main calculation, and thena version of the iterative method of Goldfarb & Idnani (1983)is applied. It is proved that the algorithm converges, the analysisbeing a special case of the theory of Goldfarb & Idnani.The algorithm is efficient because the matrices that occur arebanded due to representing the required fit as a linear combinationof B-splines. Some numerical results illustrate the method.They suggest that the algorithm can be used when n is very large,because the O(n) starting procedure identifies most of the convexityconstraints that are active at the solution.     

A potential reduction algorithm for an extended SDP problem

An extended semi-definite programming, the SDP with an additional quadratic term in the objective function, is studied. Our generalization is similar to the generalization from linear programming to quadratic programming. Optimal conditions for this new class of problems are discussed and a potential reduction algorithm for solving QSDP problems is presented. The convergence properties of this algorithm are also given.     

Shape constrained smoothing using smoothing splines

Summary  In some regression settings one would like to combine the flexibility of nonparametric smoothing with some prior knowledge about the regression curve. Such prior knowledge may come from a physical or economic theory, leading to shape constraints such as the underlying regression curve being positive, monotone, convex or concave. We propose a new method for calculating smoothing splines that fulfill these kinds of constraints. Our approach leads to a quadratic programming problem and the infinite number of constraints are replaced by a finite number of constraints that are chosen adaptively. We show that the resulting problem can be solved using the algorithm of Goldfarb and Idnani (1982, 1983) and illustrate our method on several real data sets.     

Equivalence of some quadratic programming algorithms

We formulate a general algorithm for the solution of a convex (but not strictly convex) quadratic programming problem. Conditions are given under which the iterates of the algorithm are uniquely determined. The quadratic programming algorithms of Fletcher, Gill and Murray, Best and Ritter, and van de Panne and Whinston/Dantzig are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points. The various methods are shown to differ only in the manner in which they solve the linear equations expressing the Kuhn-Tucker system for the associated equality constrained subproblems. Equivalence results have been established by Goldfarb and Djang for the positive definite Hessian case. Our analysis extends these results to the positive semi-definite case. This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A8189.     

Global optimality conditions and optimization methods for quadratic integer programming problems

In this paper, we first establish some sufficient and some necessary global optimality conditions for quadratic integer programming problems. Then we present a new local optimization method for quadratic integer programming problems according to its necessary global optimality conditions. A new global optimization method is proposed by combining its sufficient global optimality conditions, local optimization method and an auxiliary function. The numerical examples are also presented to show that the proposed optimization methods for quadratic integer programming problems are very efficient and stable.     

The Simplex and the Dual Method for Quadratic Programming

The paper presents a straightforward generalization of the Simplex and the dual method for linear programming to the case of convex quadratic programming. The two algorithms, called the Simplex and the dual method for quadratic programming, are applicable when the matrix of the quadratic part of the objective function, in case this function is to be maximized, is negative definite, negative semi-definite or zero; in the last case the two methods are equivalent to an application of the similar methods for linear programming. The paper gives an exposition of the methods as well as examples and interpretations. The relations with linear programming methods are considered and some starting procedures in case no initial feasible solution is available are presented.     

Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling

We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. This paper studies a sequential semidefinite programming (SSP) method, which is a generalization of the well-known sequential quadratic programming method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class.     

二次半定规划问题及其投影收缩算法

In this paper,we discuss the relations among the quadratic semi-definite programming problem,the linear semi-definite porgramming and the linearquadratic semi-definite programming problem.The duality theories are presented.After proving the equivalence of its optimality conditions and monotonous linear variational inequalities,we use the projection and contraction algorithms to solve(QSDP),We present the algorithms and its convergence analysis.     

Space tensor conic programming

Space tensors appear in physics and mechanics. Mathematically, they are tensors in the three-dimensional Euclidean space. In the research area of diffusion magnetic resonance imaging, convex optimization problems are formed where higher order positive semi-definite space tensors are involved. In this short paper, we investigate these problems from the viewpoint of conic linear programming (CLP). We characterize the dual cone of the positive semi-definite space tensor cone, and study the CLP formulation and the duality of positive semi-definite space tensor conic programming.     

Convergence of the BFGS-SQP Method for Degenerate Problems

In this paper, we propose a BFGS (Broyden–Fletcher–Goldfarb–Shanno)-SQP (sequential quadratic programming) method for nonlinear inequality constrained optimization. At each step, the method generates a direction by solving a quadratic programming subproblem. A good feature of this subproblem is that it is always consistent. Moreover, we propose a practical update formula for the quasi-Newton matrix. Under mild conditions, we prove the global and superlinear convergence of the method. We also present some numerical results.     

Application of sequential quadratic programming software program to an actual problem

We produced a nonlinear optimization software program which is based on a Sequential Quadratic Programming (SQP) method (Schittkowski, 1981a). Our program has several original characteristics: (1) automatic choice between two QP solvers, the Goldfarb—Idnani (GI) method (Goldfarb and Idnani, 1983) and the Least Squares (LS) method (Schittkowski, 1981b); (2) an augmented Lagrange function (Schittkowski, 1981a) as the objective function for line search; (3) adaptive Armijo method for line search; (4) direct definition of upper and lower bounds for variables and constraint functions; and (5) accurate numerical differentials. These characteristics ensure the reliability of our program for solving standard problems. In this paper, point (3) is described in detail. Then, the program is applied to an actual problem, the optimal placement of blocks. A model for this problem has been suggested by Sha and Dutton (1984), but it was unsuited to treatment by the SQP method. Thus we modify it to ensure program applicability.     

整数规划新进展

整数规划是对全部或部分决策变量为整数的最优化问题的模型、算法及应用等的研究, 是运筹学和管理科学中应用最广泛的优化模型之一. 首先简要回顾整数规划的历史和发展进程, 概述线性和非线性整数规划的一些经典方法. 然后着重讨论整数规划若干新进展, 包括0-1二次规划的半定规划~(SDP)~松弛和随机化方法, 带半连续变量和稀疏约束的优化问题的整数规划模型和方法, 以及0-1二次规划的协正锥规划表示和协正锥的层级半定规划~(SDP)~逼近. 最后, 对整数规划未来研究方向进行展望并对一些公开问题进行讨论.     

A smoothing Newton method for a type of inverse semi-definite quadratic programming problem

We consider an inverse problem arising from the semi-definite quadratic programming (SDQP) problem. We represent this problem as a cone-constrained minimization problem and its dual (denoted ISDQD) is a semismoothly differentiable (SC¹${\text{SC}}^1$) convex programming problem with fewer variables than the original one. The Karush–Kuhn–Tucker conditions of the dual problem (ISDQD) can be formulated as a system of semismooth equations which involves the projection onto the cone of positive semi-definite matrices. A smoothing Newton method is given for getting a Karush–Kuhn–Tucker point of ISDQD. The proposed method needs to compute the directional derivative of the smoothing projector at the corresponding point and to solve one linear system per iteration. The quadratic convergence of the smoothing Newton method is proved under a suitable condition. Numerical experiments are reported to show that the smoothing Newton method is very effective for solving this type of inverse quadratic programming problems.     

A note on an implementation of a method for quadratic semi-infinite programming

For convex quadratic semi-infinite programming problems aFortran-package is described. A first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems. This enables an efficient treatment of large problems with moderate storage requirements. Powell's (1983) numerically stable convex quadratic programming implementation is used to solve the QP-subproblems.     

A new local and global optimization method for mixed integer quadratic programming problems

In this paper, a new local optimization method for mixed integer quadratic programming problems with box constraints is presented by using its necessary global optimality conditions. Then a new global optimization method by combining its sufficient global optimality conditions and an auxiliary function is proposed. Some numerical examples are also presented to show that the proposed optimization methods for mixed integer quadratic programming problems with box constraints are very efficient and stable.     

交替方向法求解带线性约束的变分不等式

１引言变分不等式是一个有广泛应用的数学问题,它的一般形式是：确定一个向量,使其满足这里ｆ是一个从到自身的一个映射,Ｓ是Ｒ中的一个闭凸集．在许多实际问题中集合Ｓ往往具有如下结构其中ＡｂＫ是中的一个简单闭凸集．例如一个正卦限,一个框形约束结构,或者一个球简言之,Ｓ是Ｒ中的一个超平面与一个简单闭凸集的交．求解问题（１）－（２）,往往是通过对线性约束Ａ引人Ｌａｇｒａｎｇｅ乘子,将原问题化为如下的变分不等式：确定使得我们记问题（３）－（４）为ＶＩ（Ｆ）．熟知［３］,ＶＩ（,Ｆ）等价于投影方程其中凡（·）表…     

Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations

We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S. Zhang for a subclass of quadratic maximization problems that have nonnegative off-diagonal coefficient matrices of quadratic objective functions and diagonal coefficient matrices of quadratic constraint functions. A new SOCP relaxation is proposed for the class of nonconvex quadratic optimization problems by extracting valid quadratic inequalities for positive semidefinite cones. Its effectiveness to obtain optimal values is shown to be the same as the SDP relaxation theoretically. Numerical results are presented to demonstrate that the SOCP relaxation is much more efficient than the SDP relaxation.     

A numerically stable dual method for solving strictly convex quadratic programs

An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point. Its implementation utilizes the Cholesky and QR factorizations and procedures for updating them. The performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code (the principal motivation for the development of the algorithm). These computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available. The algorithm is also compared theoretically to the modified-simplex type dual methods of Lemke and Van de Panne and Whinston and it is illustrated by a numerical example. This research was supported in part by the Army Research Office under Grant No. DAAG 29-77-G-0114 and in part by the National Science Foundation under Grant No. MCS-6006065.     

Matrix factorizations in optimization of nonlinear functions subject to linear constraints

Several ways of implementing methods for solving nonlinear optimization problems involving linear inequality and equality constraints using numerically stable matrix factorizations are described. The methods considered all follow an active constraint set approach and include quadratic programming, variable metric, and modified Newton methods.Part of this work was performed while the author was a visitor at Stanford University. This research was supported in part by the National Science Foundation under Grant GJ 36472 and in part by the Atomic Energy Commission Contract No. AT(04-3)-326PA30.