Non-separable Objective Function Research Articles

A Stochastic Model of the Dynamic Vehicle Allocation Problem

The stochastic vehicle allocation problem addresses the movement of vehicles between locations over a given planning horizon. The demand for vehicles to carry loads between locations is uncertain, and vehicles are assumed to be able to handle several loads over the course of the planning horizon. This requires tracking the movement of both loaded and empty vehicles, resulting in a network with stochastic flows. The methodology represents the flows of vehicles over the network explicitly as random variables, taking advantage of the acyclic structure of the time space network. The decision variables are formulated in terms of sending a certain fraction of the supply of vehicles at a node (which is random) over each of the outbound links. The result is a nonseparable objective function with a very simple constraint structure which lends itself readily to the Frank-Wolfe algorithm. Numerical experiments suggest very good computational efficiency.

Optimal Power Flow By Newton Approach

The classical optimal power flow problem with a nonseparable objective function can be solved by an explicit Newton approach. Efficient, robust solutions can be obtained for problems of any practical size or kind. Solution effort is approximately proportional to network size, and is relatively independent of the number of controls or binding inequalities. The key idea is a direct simultaneous solution for all of the unknowns in the Lagrangian function on each iteration. Each iteration minimizes a quadratic approximation of the Lagrangian. For any given set of binding constraints the process converges to the Kuhn-Tucker conditions in a few iterations. The challenge in algorithm development is to efficiently identify the binding inequalities.

Technical Note—Further Results on the Minimization of a Nonseparable Objective Function Subject to Disjoint Constraints

We consider the class of nonseparable mathematical programs in which the vector variable can be partitioned into a number of subvectors corresponding to independent constraint sets, as opposed to two such subvectors considered in an earlier paper by Wendell and Hurter (W-H). Several optimality conditions previously given by W-H are extended to this more general case.

Minimization of a Non-Separable Objective Function Subject to Disjoint Constraints

We consider an important class of mathematical programs, in which the vector variable can be partitioned into two subvectors corresponding to independent constraint sets. Necessary and sufficient conditions for optimal solutions are developed, and two approaches for obtaining solutions are reviewed. We present an enumeration approach, reducing the problem to a finite number of subproblems, and show that duality makes the solution of many of the subproblems unnecessary. Next, we develop an alternating approach, wherein the problem is solved for one of the subvectors while the other is held constant, and then the subvector roles are reversed. This procedure is observed to converge to partial optimum solutions. A widely applicable subclass of problems includes a linear program in one of the subvectors. For this subclass a sufficient condition for local optimality is determined. The condition is easily testable and fails to hold, in many cases, only if a better solution is obtained. Also, this condition shows that partial optimum solutions are almost always local optima.