Class Of Interior-point Algorithms Research Articles

On the use of a class of interior point algorithms in stochastic structural optimization

In this paper the feasibility of using a particular feasible direction interior point algorithm for solving reliability-based optimization problems of high dimensional stochastic dynamical systems is investigated. The optimal design problem is formulated in terms of an inequality constrained non-linear optimization problem. A class of interior point algorithms based on the solution of the first-order optimality conditions is considered here. For this purpose, a quasi-Newton iteration is used to solve the corresponding nonlinear system of equations. Several numerical examples are presented to illustrate the feasibility of the proposed methodology.

Interior point methods in DEA to determine non-zero multiplier weights

Multiplier weights in DEA are obtained by solving any one of several multiplier linear programs (LPs). These weights are a fundamental aspect of DEA and have many uses and interpretations including determining marginal rates of substitution. Obtaining usable values from multiplier weights can be problematic due to the way DEA LPs are formulated and solved. For example, extreme efficient DMUs generate multiple optima in standard multiplier LPs. This is also true of any efficient or virtual DMU on a face of the production possibility set with less than full dimension. Another problem arises when the LP generates optimal solutions where one or more of the multipliers are zero. An important class of interior point algorithms for LP known as “path-following” methods addresses these two issues about finding optimal multiplier weights in DEA: (1) reproducibility, that is, the optimal solution is independent of a starting point since it is generated by applying a well-defined optimization criterion; (2) non-zero multipliers, whereby the multiplier weights associated with an optimal solution for a point in the efficient frontier are never zero. In the process of exploring these methods for DEA we introduce the “multiplier generator” DEA LP formulated to provide access to all multiplier vectors for points on the efficient frontier. Our results provide prescriptions and recommendations for using path-following solvers in DEA.Scope and purpose. This is a study on the generation of non-zero weights in DEA using interior point methods. The purpose is to generate these weights efficiently and in a manner that can be replicated independently. There is a clear demand for non-zero multiplier weights in DEA for use to price the attributes. We introduce new LP formulations specifically designed to provide access to the full set of multiplier weights at the DMU being scored.

Dual interior point algorithms

For a linear programming problem stated in the canonical form we consider the dual problem and describe a class of interior point algorithms which generate monotonically improving approximations to its solution. We theoretically substantiate the optimization process in the admissible domain under the assumption that the dual problem is nondegenerate. In addition, we describe subsets of algorithms that lead to relative interior points of optimal solutions. These algorithms have linear and superlinear convergence rates. Moreover, we obtain a special subset of algorithms which generate iterative sequences of approximations to a solution of the direct problem, whose convergence rate exceeds that of the sequences of monotonically improving approximations to a solution of the dual problem.

On a class of interior point algorithms

A family of interior point algorithms for solving linear programs is examined. Under the assumption on the nondegeneracy of the problem, a theoretical justification of these algorithms is given. The sets of the algorithms converging to relatively interior optimal solutions and having linear or superlinear convergence rate are identified.

Mathematical programs with equilibrium constraints (MPECs) in process engineering

Mathematical programs with equilibrium constraints (MPECs) form a relatively new and interesting subclass of nonlinear programming problems. In this paper we propose a novel method of solving MPECs by appropriate reformulation of the equilibrium conditions. The reformulation can be easily incorporated in a certain class of interior point algorithms for nonlinear optimization. The algorithm used in the study follows a primal-dual interior point approach and shows encouraging results on a test suite of MPECs. The algorithm is also able to perform optimization of distillation columns with phase changes and tray optimization using only continuous variables. We also consider a number of topics to improve performance of the algorithm and to identify classes of process engineering problems that can be handled as MPECs.

On the computer implementation of feasible direction interior point algorithms for nonlinear optimization

The paper discusses the computer implementation of a class of interior point algorithms for the minimization of nonlinear functions with equality and inequality constraints. These algorithms consist of fixed point iterations to solve KKT firstorder optimality conditions. At each iteration a descent direction is defined by solving a linear system. Then, the linear system is perturbed in such a way as to deflect the descent direction and obtain a feasible descent direction. A line search is finally performed to obtain a new interior point with a lower objective. Newton, quasi-Newton, or first-order versions of the algorithm can be obtained. This paper is mainly concerned with the solution of the internal linear systems, the algorithms that are employed for the constrained line search and also with the quasi-Newton matrix updating. Some numerical results obtained with a quasi Newton algorithm are also presented. A set of test problems were solved very efficiently with the same values of the internal parameters.

Newton-type Methods with Generalized Distances For Constrained Optimization

We consider a class of interior point algorithms for minimizing a twice continuously differentiable function over a closed convex set with nonempty interior. On one hand, our algirothms can be viewed as an approximate version of the generalized proximal point methods and, on the other hand, as an extension of unconstrained Newton-type methods to the constrained case. Each step consists of solving a strongly convex unconstrained program followed by a one-dimensional search along either a line or a curve segment in the interior of the feasible set. The information about the feasible set is contained in the generalized distance function whose gradient diverges on the boundary of this set. When the feasible set is the whole space, the standard regularized Newton method is a particular case in our framework. We show, under standard assumptions, that every accumulation point of the sequence of iterates satisfies a first order necessary optimality condition for the problem and solves the problem if the objective function is convex. Some computational results are also reported