Algorithms for solving a class of nonconvex optimization problems. Methods of subgradients. (English) Zbl 0638.90087

Fermat days 85: Mathematics for optimization, Toulouse/France 1985, North-Holland Math. Stud. 129, 249-271 (1986).
[For the entire collection see Zbl 0595.00017.]
The authors are concerned with the following optimization problem \[ \text{Given }\lambda =\sup\{f(x)| x\in C\}\text{ find } x^* \text{ such that }f(x^*)= \lambda, \tag{P} \] where \(f: R^ n\to R\) is convex and C is a nonempty closed convex set.
Investigation strictly related to the present one have been carried out by the first author in several papers [see, Numer. Math. 45, 377-401 (1984; Zbl 0531.65022), and Linear Algebra Appl. 62, 163-182 (1984; Zbl 0563.65029)].
Here, first duality schemes for (P) are investigated then analytical characterizations of the optimal solutions are given. Such results lead to develop various methods based on subgradients for finding the optimal solutions of (P). Their convergence properties are also investigated.
Reviewer: M.Gaviano


90C30 Nonlinear programming
49M37 Numerical methods based on nonlinear programming
65K05 Numerical mathematical programming methods
49N15 Duality theory (optimization)