Generalized monotone line search SQP algorithm for constrained minimax problems

A proximal-projection partial bundle method for convex constrained minimax problems

“…Due to the special structure of f (x), general-purpose nonsmooth methods ( [33,22,3]) may not be efficient to solve them. So, by exploiting their peculiar structure, a great deal of effort has been devoted to developing efficient methods for minimax problems, see, e.g., [37,6,31,23,41,25,38,39,9,14,15,10,11,7,12].…”

“…In particular, for the case where f i (i ∈ I) are continuously differentiable, one common approach for solving the original problem is the smoothing method. There are mainly two smoothing approaches: (i) by introducing a smoothing function, the original problem is approximated by a sequence of smooth optimization problems, and the solution can be obtained from the smoothing function, see e.g., [23,41]; (ii) by introducing an auxiliary variable, the minimax problem can be equivalently reformulated as a smooth problem, and then smooth methods can be designed to solve this special smooth problem, such as penalty methods [6], trust region methods [37,38,39], and sequential (quadratically constrained) quadratic programming [31,14,15].…”

“…• (On Step 7) Once the algorithm enters Step 7, the feasible descent property (see Lemma 2.2 below) is satisfied, and therefore we accept y k+1 as a new stability center. Note that the component function values f i (y k+1 ) for all i ∈ I are already calculated whenever the algorithm enters Step 7, so we can simply use (14) to enrich the bundle without extra computational cost.…”

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

“…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.)…”

Improved Filter-SQP Algorithm with Active Set for Constrained Minimax Problems

“…However, it is well-known that the solution of (8) may not be a feasible descent direction and can not avoid the Maratos effect. Recently, many researches have extended the popular SQP scheme to the minimax problems (see [21][22][23][24][25], etc.). Jian et al [22] and Hu et al [23] process pivoting operation to generate an -active constraint subset associated with the current iteration point.…”