Nonlinear programming

About: Nonlinear programming is a research topic. Over the lifetime, 19486 publications have been published within this topic receiving 656602 citations. The topic is also known as: non-linear programming & NLP.

A combined trip generation, trip distribution, modal split, and trip assignment model. in: the automobile

Abstract: The goal of this paper was to develop a model and a computationally efficient solution procedure that would achieve a practical compromise between the two objectives of behavioral richness and computational tractability. In the model developed, trip generation depends upon the transportation system's performance through an accessibility measure that is based on the random utility theory of users' behavior. In addition, trip distribution is given by a logit model rather than being specified by a more rigid entropy model. The results of this study indicate that assumptions of invertability, symmetry, and negative definiteness of transportation demand functions are not as restrictive as previously thought. Therefore, the reformulation of traffic equilibrium models as equivalent optimization problems that can be solved efficiently by nonlinear programming methods remains a potentially fruitful avenue for future investigation.

First Order Conditions for General Nonlinear Optimization

Abstract: We show how first order optimality conditions for a very general nonlinear optimization problem may be derived in a conceptually simple and unified manner in terms of certain multivalued functions associated with the problem. Necessary conditions for general problems and sufficient conditions for convex problems are developed, and the classical multiplier conditions are shown to be related in a simple way to these.

Solving nonlinear multicommodity flow problems by the analytic center cutting plane method

Abstract: The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with Dijkstra's d-heap algorithm. An implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on well-known nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities).

Directional derivatives of the solution of a parametric nonlinear program

Abstract: Consider a parametric nonlinear optimization problem subject to equality and inequality constraints. Conditions under which a locally optimal solution exists and depends in a continuous way on the parameter are well known. We show, under the additional assumption of constant rank of the active constraint gradients, that the optimal solution is actually piecewise smooth, hence B-differentiable. We show, for the first time to our knowledge, a practical application of quadratic programming to calculate the directional derivative in the case when the optimal multipliers are not unique.