A Nonlinear Lagrange Algorithm for Minimax Problems with General Constraints

An Implementable SAA Nonlinear Lagrange Algorithm for Constrained Minimax Stochastic Optimization Problems

“…If assumptions (d)-(h) are satisfied, then it follows from Theorem 3.1 of [13] that there exist > 0 ( < ) and̂∈ (0, 1) such that as → ∞, ( ) → * , ( ) → * , V ( ) → V * , ( ) → * , and ( ) → * for any ( (0) , V (0) , (0) , (0) , ) ∈ ( * , V * , * , * ) × (0,̂). If assumptions (a)-(c) are satisfied, and̂( 0) , = (0) ( = 1, .…”

“…The Lagrange function for problem (1) is defined by ( , , V, ) = ∑ ∈ ( ) + ∑ ∈ V ( ) + ∑ ∈ ( ). Let ( * , * , V * , * ) denote the Karush-Kuhn-Tucker (KKT) solution of problem (1) and * = ( * ) (see [13]). Let > 0 be small enough.…”

“…On the one hand, as one of the famous smoothing techniques aiming to overcome the nonsmoothness of ( ) (see [1,[8][9][10][11][12][13][14][15][16]), the nonlinear Lagrange method has many interesting merits, such as no restrictions on the feasibility of variables , improvement on the convergence rate and the numerical robustness compared with penalty method by introducing the Lagrangian multipliers as the main driving force. On the other hand, the sample average approximation method is one of the well-behaved approaches for solving stochastic programming problems, the basic idea of which is to generate an independent and identically distributed (i.i.d.)…”

“…Motivated by the effectiveness of the nonlinear Lagrange method, this paper presents a nonlinear Lagrange function for problem (1) based on the work in [13] as follows: . .…”

“…, }, > 0 is a penalty parameter, and ∈ R is an estimate of the objective function ( ). ( , , V, , , ) has good properties and the corresponding nonlinear Lagrangian algorithm is recalled below (see [13]).…”