An Algorithm for Non-Linear Network Programming: Implementation,Results and Comparisons

This paper evaluates an algorithm for solving network flow optimization problems with quadratic cost functions. Strategies for fast implementation are discussed and the results of extensive numerical tests are given. The performance of the algorithm measured by CPU time is compared with that of the convex simplex method specialized for quadratic network programming. Performance of the two methods is analysed with respect to network size and density, and other parameters of interest. The algorithm is shown to perform significantly better on the majority of problems. We also show how the algorithm may be used to solve non-linear convex network optimization problems by the use of sequential quadratic programming.     

一个关于二次规划问题的分段线性同伦算法

本文发展了一个关于二次规划问题的分段线性同伦算法。该算法可看作是外点罚函数法的一个变体。凡是符合外点罚函数法收敛条件的二次规划问题用该算法均可经有限次轮回运算得到稳定解。大量的关于随机的凸二次规划问题的数值实验结果表明它的计算效率是高的，在某些条件下可能是多项式时间算法。     

A parameterized hessian quadratic programming problem

We present a general active set algorithm for the solution of a convex quadratic programming problem having a parametrized Hessian matrix. The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplying a symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval upon which the parametric Hessian is positive semidefinite. The algorithm is general in that any of several existing quadratic programming algorithms can be extended in a straightforward manner for the solution of the parametric Hessian problem.This research was supported by the Natural Sciences and Engineering Research Council under Grant No. A8189 and under a Postgraduate Scholarship, by an Ontario Graduate Scholarship, and by the University of Windsor Research Board under Grant No. 9432.     

A parameterized hessian quadratic programming problem

We present a general active set algorithm for the solution of a convex quadratic programming problem having a parametrized Hessian matrix. The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplying a symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval upon which the parametric Hessian is positive semidefinite. The algorithm is general in that any of several existing quadratic programming algorithms can be extended in a straightforward manner for the solution of the parametric Hessian problem. This research was supported by the Natural Sciences and Engineering Research Council under Grant No. A8189 and under a Postgraduate Scholarship, by an Ontario Graduate Scholarship, and by the University of Windsor Research Board under Grant No. 9432.     

Duality Bound Method for the General Quadratic Programming Problem with Quadratic Constraints

The purpose of this article is to develop a branch-and-bound algorithm using duality bounds for the general quadratically-constrained quadratic programming problem and having the following properties: (i) duality bounds are computed by solving ordinary linear programs; (ii) they are at least as good as the lower bounds obtained by solving relaxed problems, in which each nonconvex function is replaced by its convex envelope; (iii) standard convergence properties of branch-and-bound algorithms for nonconvex global optimization problems are guaranteed. Numerical results of preliminary computational experiments for the case of one quadratic constraint are reported.     

A very simple SQCQP method for a class of smooth convex constrained minimization problems with nice convergence results

We introduce a new and very simple algorithm for a class of smooth convex constrained minimization problems which is an iterative scheme related to sequential quadratically constrained quadratic programming methods, called sequential simple quadratic method (SSQM). The computational simplicity of SSQM, which uses first-order information, makes it suitable for large scale problems. Theoretical results under standard assumptions are given proving that the whole sequence built by the algorithm converges to a solution and becomes feasible after a finite number of iterations. When in addition the objective function is strongly convex then asymptotic linear rate of convergence is established.     

An SQP-type algorithm for nonlinear second-order cone programs

We propose an SQP-type algorithm for solving nonlinear second-order cone programming (NSOCP) problems. At every iteration, the algorithm solves a convex SOCP subproblem in which the constraints involve linear approximations of the constraint functions in the original problem and the objective function is a convex quadratic function. Those subproblems can be transformed into linear SOCP problems, for which efficient interior point solvers are available. We establish global convergence and local quadratic convergence of the algorithm under appropriate assumptions. We report numerical results to examine the effectiveness of the algorithm. This work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.     

Predictor-Corrector Algorithm for Convex Quadratic Programming with Upper Bounds

1.IntroductionThepredictor-correctormethodforlinearprogrammingisawellknownillteriorpointmethoddevelopedbyMizunoetal.[1])duetoitsquadraticaJlyconvergentanalysis.ThiskindofanalysisusuaJlycontainstwosteps,i.e.,predictorstepandcorrectorstepasoneiteration.ThecorrectorstepisusedonlytoensurethattheiteratesstayclosetothecelltraJpathsothatlargestepcanbetakenduringthepredictorstep.Thedualitygapremainsunchangedatcorrectorstepforlinearprogramming,butincaseofconvexquadraticprogramming,asshownlaterofthisp…     

A convergence proof for an affine-scaling algorithm for convex quadratic programming without nondegeneracy assumptions

This paper presents a theoretical result on convergence of a primal affine-scaling method for convex quadratic programs. It is shown that, as long as the stepsize is less than a threshold value which depends on the input data only, Ye and Tse's interior ellipsoid algorithm for convex quadratic programming is globally convergent without nondegeneracy assumptions. In addition, its local convergence rate is at least linear and the dual iterates have an ergodically convergent property.Research supported in part by the NSF under grant DDM-8721709.     

Convergence property of the Iri-Imai algorithm for some smooth convex programming problems

In this paper, the Iri-Imai algorithm for solving linear and convex quadratic programming is extended to solve some other smooth convex programming problems. The globally linear convergence rate of this extended algorithm is proved, under the condition that the objective and constraint functions satisfy a certain type of convexity, called the harmonic convexity in this paper. A characterization of this convexity condition is given. The same convexity condition was used by Mehrotra and Sun to prove the convergence of a path-following algorithm.The Iri-Imai algorithm is a natural generalization of the original Newton algorithm to constrained convex programming. Other known convergent interior-point algorithms for smooth convex programming are mainly based on the path-following approach.     

A quadratic programming algorithm

By using conjugate directions a method for solving convex quadratic programming problems is developed. The algorithm generates a sequence of feasible solutions and terminates after a finite number of iterations. Extensions of the algorithm for nonconvex and large structured quadratic programming problems are discussed.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 and in part by the Natural Sciences and Engineering Research Council of Canada under Grant Nos. A 8189 and E 5582.     

A parametric successive underestimation method for convex multiplicative programming problems

This paper addresses itself to the algorithm for minimizing the product of two nonnegative convex functions over a convex set. It is shown that the global minimum of this nonconvex problem can be obtained by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in a higher dimensional space and to apply a branch-and-bound algorithm using an underestimating function. Computational results indicate that our algorithm is efficient when the objective function is the product of a linear and a quadratic functions and the constraints are linear. An extension of our algorithm for minimizing the sum of a convex function and a product of two convex functions is also discussed.     

Extensions of Dinkelbach's algorithm for solving non-linear fractional programming problems

Dinkelbach's algorithm was developed to solve convex fractinal programming. This method achieves the optimal solution of the optimisation problem by means of solving a sequence of non-linear convex programming subproblems defined by a parameter. In this paper it is shown that Dinkelbach's algorithm can be used to solve general fractional programming. The applicability of the algorithm will depend on the possibility of solving the subproblems. Dinkelbach's extended algorithm is a framework to describe several algorithms which have been proposed to solve linear fractional programming, integer linear fractional programming, convex fractional programming and to generate new algorithms. The applicability of new cases as nondifferentiable fractional programming and quadratic fractional programming has been studied. We have proposed two modifications to improve the speed-up of Dinkelbachs algorithm. One is to use interpolation formulae to update the parameter which defined the subproblem and another truncates the solution of the suproblem. We give sufficient conditions for the convergence of these modifications. Computational experiments in linear fractional programming, integer linear fractional programming and non-linear fractional programming to evaluate the efficiency of these methods have been carried out.     

A feasible filter SQP algorithm with global and local convergence

A feasible sequential quadratic programming (SQP) filter algorithm is proposed for general nonlinear programming. It is based on the modified quadratic programming (QP) subproblem in which each iteration proceeds in two phases. The first phase solves a general convex QP problem which does not require any feasibility restoration phase whose computation may be expensive. And, under some mild conditions, the global convergence is proved. The second phase can make the presented SQP method derive quadratic convergence by employing exact Hessian information.     

A POLYNOMIAL PREDICTOR-CORRECTOR INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one corrector step after each predictor step, where Step 2 is a predictor step and Step 4 is a corrector step in the algorithm. In the algorithm, the predictor step decreases the dual gap as much as possible in a wider neighborhood of the central path and the corrector step draws iteration points back to a narrower neighborhood and make a reduction for the dual gap. It is shown that the algorithm has O(n~(1/2)L) iteration complexity which is the best result for convex quadratic programming so far.     

A method for minimizing the sum of a convex function and a continuously differentiable function

This paper presents a method for minimizing the sum of a possibly nonsmooth convex function and a continuously differentiable function. As in the convex case developed by the author, the algorithm is a descent method which generates successive search directions by solving quadratic programming subproblems. An inexact line search ensures global convergence of the method to stationary points.     

Logarithmic SUMT limits in convex programming

The limits of a class of primal and dual solution trajectories associated with the Sequential Unconstrained Minimization Technique (SUMT) are investigated for convex programming problems with non-unique optima. Logarithmic barrier terms are assumed. For linear programming problems, such limits – of both primal and dual trajectories – are strongly optimal, strictly complementary, and can be characterized as analytic centers of, loosely speaking, optimality regions. Examples are given, which show that those results do not hold in general for convex programming problems. If the latter are weakly analytic (Bank et al. [3]), primal trajectory limits can be characterized in analogy to the linear programming case and without assuming differentiability. That class of programming problems contains faithfully convex, linear, and convex quadratic programming problems as strict subsets. In the differential case, dual trajectory limits can be characterized similarly, albeit under different conditions, one of which suffices for strict complementarity. Received: November 13, 1997 / Accepted: February 17, 1999?Published online February 22, 2001     

关于二次规划问题分段线性同伦算法的改进

本文利用Ｃｈｏｌｅｓｋｙ分解，Ｇａｕｓｓ消去等技术和定义适当的同伦映射，将关于二次规划问题的分段线性同伦算法加以改进，改进后的算法，对于严格凸二次规划来说，计算效率与Ｇｏｌｄｆａｒｂ－Ｉｄｎａｎｉ的对偶法相当。     

A new branching rule for the branch and bound algorithm for solving nonlinear integer programming problems

We examine a branch and bound algorithm for solving nonlinear (convex) integer programming problems. In this note we generalize previous results for the quadratic case. The variables are branched in such a way that the number of branch and bound nodes checked in the process is small. Numerical results confirm the efficiency.     

More with the Lemke complementarity algorithm

In the case that the matrix of a linear complementarity problem consists of the sum of a positive semi-definite matrix and a co-positive matrix a general condition is deduced implying that the Lemke algorithm will terminate with a complementarity solution. Applications are presented on bi-matrix games, convex quadratic programming and multi-period programs.Contributed to the XXIII TIMS Meeting, Athens, July 1977.