Generalised monotone line search algorithm for degenerate nonlinear minimax problems

Abstract

In this paper, nonlinear minimax problems are discussed. Using Sequential Quadratic Programming and the generalised monotone line search technique, we propose a new algorithm for solving degenerate minimax problems. At each iteration of the proposed algorithm, a search direction is obtained by solving a new Quadratic Programming problem which always has a solution. Global convergence can be obtained without the regularity condition of linear independence. Finally, some numerical experiments are reported.

A sequential quadratically constrained quadratic programming method for unconstrained minimax problems

“…So, many authors have applied the idea of SQP method to present effective algorithms for solving the minimax problems, such as in Refs. [4][5][6][7][8][9][10][11][12]. It is a key problem of various SQP methods to overcome the so-called Maratos effect [13] under suitable conditions, for example, to solve one or more additional quadratic programs or systems of linear equations, or compute explicit correction directions.…”

A QP-Free Algorithm for Finite Minimax Problems

“…The computational results are reported in Table 1, and the columns of Table 1 have the following meanings: IP: the initial point; : the number of variables; : the number of functions ( ); ALG: the type of algorithm; NI: the number of iterations. "Algo A" represents Algorithm A in this paper, "J2006-1" and "J2006-2" represent the algorithms in [13], and "Hu2009" represents the algorithm in [15].…”

“…From Table 1, we can see that our algorithm can find the solutions of the test problems with a small number of iterations, and the computational results illustrate that our algorithm executes well for those problems. The numerical results are comparative with the algorithms in [13,15]. Furthermore, we only need to solve two systems of linear equations with the same coefficient matrix per iteration.…”

“…The third one is based on solving the equivalent smooth nonlinear programming problem (3), such as the sequential quadratic programming (SQP) methods (see [11][12][13][14][15][16]), the trust-region strategies (see [17,18]), sequential quadratically constrained quadratic programming (SQCQP) method (see [19]), gradient projection method (see [20]), and the interior point (IP) methods (see [21][22][23]). The advantage of this class method is that the nondifferentiable optimization problem (1) can be transformed into an equivalent smooth constrained nonlinear programming problem which can be solved by well-established methods.…”