An approach to the solution of nonlinear unconstrained optimization problems
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BRIEF COMMUNICATIONS AN APPROACH TO THE SOLUTION OF NONLINEAR UNCONSTRAINED OPTIMIZATION PROBLEMS
UDC 519.8
Yu. P. Laptin
An approach to reducing a constrained convex programming problem to an unconstrained optimization problem is considered. An initial internal feasible point is supposed to be specified. An equivalent unconstrained optimization problem is formulated in such a way that the calculated values of gradients (subgradients) of original functions do not violate the initial constraints. Properties of introduced functions are investigated. Convexity conditions are formulated for the unconstrained optimization problem. The results may by useful for the development of algorithms for solving constrained optimization problems. Keywords: convex programming, unconstrained optimization, conditional e-subdifferential, almost-differentiable function.
In solving nonlinear constrained optimization problems, the formation of an auxiliary unconstrained optimization problem whose solution coincides with the solution of the original problem is one of frequently used expedients. Penalty function methods, in particular, the method of nonsmooth penalty functions that is widely used in convex programming [1–4], are among such expedients. In using these methods, definite difficulties arise that are connected with the selection of values of penalty coefficients. Additional problems arise in the case when functions describing a problem are defined on bounded sets [5]. In this paper, a possible approach to the reduction of a convex programming problem to an unconstrained optimization problem is considered. The initial point belonging to the interior of a feasible set is supposed to be given. An equivalent unconstrained optimization problem is formed in such a manner that the gradients (subgradients) and values of functions of the original problem are computed only at points of the feasible set. The formulated problem turns out to be closely connected with the concept of a conditional e-subdifferential [3, 4]. Properties of introduced functions are investigated. Conditions are formulated under which an unconstrained optimization problem turns out to be convex. The obtained results can be useful in developing algorithms for solution of constrained optimization problems. 1. We consider the following convex programming problem: find f * = min f ( x )
(1)
h( x ) £ 0 ,
(2)
under the constraints
where x Î R n and f , h : R n ® R are convex functions assuming finite values for any x. We denote S = { x Î R n : h ( x ) £ 0}. A feasible point x 0 Î S such that we have h( x 0 ) < 0 is assumed to be given. Cybernetics Institute, National Academy of Sciences of Ukraine, Kiev, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 182–187, May–June 2009. Original article submitted November 20, 2008. 1060-0396/09/4503-0497
©
2009 Springer Science+Business Media, Inc.
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For an arbitrary point x Î R n , we define the following mapping onto the set S : ~ : (1- a ~) x0 + a ~x Î S , 0 £ a ~ £ 1}. p S
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