On certain optimization methods with finite-step inner algorithms for convex finite-dimensional problems with inequality constraints

I. P. Antipin, A. Z. Ishmukhametov, Yu. G. Karyukina

Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119991, Russia

Abstract: Numerical methods are proposed for solving finite-dimensional convex problems with inequality constraints satisfying the Slater condition. A method based on solving the dual to the original regularized problem is proposed and justified for problems having a strictly uniformly convex sum of the objective function and the constraint functions. Conditions for the convergence of this method are derived, and convergence rate estimates are obtained for convergence with respect to the functional, convergence with respect to the argument to the set of optimizers, and convergence to the $g$-normal solution. For more general convex finite-dimensional minimization problems with inequality constraints, two methods with finite-step inner algorithms are proposed. The methods are based on the projected gradient and conditional gradient algorithms. The paper is focused on finite-dimensional problems obtained by approximating infinite-dimensional problems, in particular, optimal control problems for systems with lumped or distributed parameters.

Key words: convex finite-dimensional optimization problems, inequality constraints, numerical optimization algorithms, regularization methods.

UDC: 519.658

Received: 06.05.2006
Revised: 26.04.2007

 English version: Computational Mathematics and Mathematical Physics, 2007, 47:12, 1928–1937

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