Certain questions in solving block nonlinear optimization problems with coupling variables

Abstract

A decomposition scheme for block nonlinear convex programming problems with coupling variables is considered. The possibility is examined of using approximated solutions of subproblems for generating subgradients of the functions that appear in the master problem. A regularization of the original problem that simplifies the master problem is considered.In the paper, we consider block nonlinear convex programming problems with coupling variables, which originate, in particular, in modeling complex engineering systems [1]. An efficient approach to solving such problems is decomposition schemes, where the original complex problem is reduced to a set of more simple subproblems for each block and a special master problem. In solving the latter, subproblems for blocks are solved at each iteration.The use of decomposition schemes leads to natural paralleling of optimization algorithms and allows applying such algorithms in multiprocessing systems.The properties of the functions appearing in the master problem were analyzed in different studies, in particular, in [2]. It was assumed there that the subproblem for each block is solved exactly. The papers [3-5] deal with approximate solutions of subproblems in decomposition schemes. The approach proposed there is based on the solution of an approximation linear programming problem, has a clear geometrical meaning, but appears computationally unstable for some classes of problems [5]. We analyze here the possibilities of using approximate solutions and the information generated by e-subgradient methods [6-8] and by linearization methods [9, 10] during the solution of subproblems.The functions appearing in the master problem are generally defined on bounded sets given implicitly. This complicates the use of optimization methods. Laptin [3] considered regularization of the original problem where the functions of the master problem are defined for any values of coupling variables. The approach proposed is based on introducing auxiliary variables and leads to doubling the number of variables for each subproblem. The regularization proposed here is based on introducing one auxiliary variable for each subproblem.

An approach to the solution of nonlinear unconstrained optimization problems

“…The one-dimensional search for the value of a can be realized rather efficiently. A close approach was used in [6] for regularizing problems in schemes of decomposition with respect to variables.…”