Methods For Convex Quadratic Programming Research Articles

A Long-step barrier method for convex quadratic programming

In this paper we propose a long-step logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along projected Newton directions are taken until the iterate is in the vicinity of the center associated with the current value of the barrier parameter. We prove that the total number of iterations isO(źnL) orO(nL), depending on how the barrier parameter is updated.

An Interior-Point Algorithm for Linearly Constrained Optimization

This paper describes an algorithm for optimization of a smooth function subject to general linear constraints. An algorithm of the gradient projection class is used, with the important feature that the “projection” at each iteration is performed by using a primal–dual interior-point method for convex quadratic programming. Convergence properties can be maintained even if the projection is done inexactly in a well-defined way. Higher-order derivative information on the manifold defined by the apparently active constraints can be used to increase the rate of local convergence.

On an instance of the inverse shortest paths problem

The inverse shortest paths problem in a graph is considered, that is, the problem of recovering the arc costs given some information about the shortest paths in the graph. The problem is first motivated by some practical examples arising from applications. An algorithm based on the Goldfarb-Idnani method for convex quadratic programming is then proposed and analyzed for one of the instances of the problem. Preliminary numerical results are reported.

A Polynomial Method of Weighted Centers for Convex Quadratic Programming

Abstract A generalization of the weighted central path-following method for convex quadratic programming is presented. This is done by uniting and modifying the main ideas of the weighted central path following method for linear programming and the interior point methods for convex quadratic programming. By means of the linear approximation of the weighted logarithmic barrier function and weighted inscribed ellipsoids, ‘weighted’ trajectories are defined. Each strictly feasible primal-dual point pair defines such a weighted trajectory. The algorithm can start in any strictly feasible point that defines a weighted trajectory, which is followed through the algorithm. This algorithm has the nice feature, that it is not necessary to start the algorithm close to the central path and so additional transformations are not needed. In return, the theoretical complexity of our algorithm is dependent on the position of the starting point. Polynomiality is proved under the usual mild conditions.

An O(n 3 L) primal interior point algorithm for convex quadratic programming

We present a primal interior point method for convex quadratic programming which is based upon a logarithmic barrier function approach. This approach generates a sequence of problems, each of which is approximately solved by taking a single Newton step. It is shown that the method requires $$O(\sqrt n L)$$ iterations and O(n3.5L) arithmetic operations. By using modified Newton steps the number of arithmetic operations required by the algorithm can be reduced to O(n3L).

A STOCHASTIC PROGRAMMING MODEL FOR AGRICULTURAL PLANNING UNDER UNCERTAIN SUPPLY-DEMAND RELATIONS

under uncertain supply-demand relations and to propose an algorithm for the model. We first illustrate that several economic problems under undertaint~ must be formulated as a quadratic programming model in which the linear coefficients are stochastic variables. The model is considered as an extension of the linear stochastic programming model reported by Freund (1956) and Kataoka (1963, 1967). The following alternative criteria for optimization are introduced to complete the model: (1) maximizing the expected value of utility; (2) maximizing aspiration level; (3) maximizing probability. The deterministic equiva­ lents are then derived with the mathematical characteristics of the solutions. The equivalence relations among the eqUivalents are also clarified. We propose an iteration algorithm for the equivalents by taking advantage of the relations. The algorithm is based on lhe secant method and the convex quadratic programming method. Finally, an application of the model is illustrated with a procedure for application.

Optimal design of FIR linear phase digital filters via convex quadratic programming method

Abstract Many methods exist for designing FIR linear phase digital filters by using various optimization techniques. An optimization design method via the convex quadratic programming method is investigated in this paper. The aim of this method is to design digital filters with certain desired frequency responses that keep the passband ripple to a minimum while maintaining a balance between opposite characteristics of frequency responses, i.e. the narrow transition band and the great stopband attenuation. In order to verify the effectiveness of this method, low-pass filters are designed and comparison studies with other typical optimization methods are given.

A weighted gram-schmidt method for convex quadratic programming

Range-space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases. In this paper we describe a range-space method based upon updating a weighted Gram-Schmidt factorization of the constraints in the active set. The updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active-set strategy. Many quadratic programming problems include simple bounds on all the variables as well as general linear constraints. A feature of the proposed method is that it is able to exploit the structure of simple bound constraints. This allows the method to retain efficiency when the number ofgeneral constraints active at the solution is small. Furthermore, the efficiency of the method improves as the number of active bound constraints increases.

A constrained load flow using convex structure (modified algorithm)

This paper presents a method to solve constrained load-flow problems in which the constraint domained is determined by upper and lower bounds of active power, reactive power, and voltage. Three methods are presented that attempt to find convergent solutions to load-flow problems. The three methods are: 1. linear programming method, 2. convex quadratic programming method, 3. mixed method. For the mixed programming method, the linear programming method is used first, followed by convex quadratic programming if the convergence speed of the linear programming method has slowed down. The proposed methods are based on the simplex method which is used widely for linear programming and quadratic programming. The validity of the proposed methods are verified by numerical examples and the results obtained by the proposed methods are compared with those by the Newton-Raphson method.