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**Nonsmooth approach to optimization problems with equilibrium constraints. Theory, applications and numerical results.**
*(English)*
Zbl 0947.90093

Nonconvex Optimization and Its Applications. 28. Dordrecht: Kluwer Academic Publishers. xxi, 273 p. Dfl 240.00; $ 129.00; £82.00 (1998).

The present monogragh grew out of a series of papers on mathematical programs with equilibrium constraints published by the authors in the years 1993-1997. It is focused on the following optimization problem:

minimize \(f(x,z)\) subject to \(z\in S(x)\) nad \(x\in U_{ad}\).

Here \(f\) maps \(\mathbb{R}^n \times\mathbb{R}^k\) into \(\mathbb{R}\), \(U_{ad}\) is a nonempty closed subset of \(\mathbb{R}^n\), and \(S\) is the solution map of the perturbed generalized equation \[ 0\in C(x,z)+ N_Q(z), \] where \(C:\mathbb{R}^n\times \mathbb{R}^k \to\mathbb{R}^k\) is a continuous map, \(Q\) is a nonempty closed convex subset of \(\mathbb{R}^k\), and \(N_Q (z)\) is the normal cone to \(Q\) at \(z\in\mathbb{R}^k\). The relation \(z\in S(x)\) is called equilibrium constraint. Supposing that this constraint locally defines a certain implicit function, the authors convert the considered problem into an “easier” mathematical programming problem with a nonsmooth objective function. By applying the “non-differentiable” calculus of Clarke to the new problem, they derive necessary optimality conditions for the initial problem. Furthermore, by using the bundle method of nonsmooth optimization for the new problem, they get a solution technique for the initial problem. The efficiency of this approach is shown on a series of tough nonacademic problems from the area of optimum shape design and economic modelling.

Dealing with both, the theory and applications of optimization problems with equilibrium constraints, this book is extremely useful for researchers in optimization theory and for applied mathematicians.

minimize \(f(x,z)\) subject to \(z\in S(x)\) nad \(x\in U_{ad}\).

Here \(f\) maps \(\mathbb{R}^n \times\mathbb{R}^k\) into \(\mathbb{R}\), \(U_{ad}\) is a nonempty closed subset of \(\mathbb{R}^n\), and \(S\) is the solution map of the perturbed generalized equation \[ 0\in C(x,z)+ N_Q(z), \] where \(C:\mathbb{R}^n\times \mathbb{R}^k \to\mathbb{R}^k\) is a continuous map, \(Q\) is a nonempty closed convex subset of \(\mathbb{R}^k\), and \(N_Q (z)\) is the normal cone to \(Q\) at \(z\in\mathbb{R}^k\). The relation \(z\in S(x)\) is called equilibrium constraint. Supposing that this constraint locally defines a certain implicit function, the authors convert the considered problem into an “easier” mathematical programming problem with a nonsmooth objective function. By applying the “non-differentiable” calculus of Clarke to the new problem, they derive necessary optimality conditions for the initial problem. Furthermore, by using the bundle method of nonsmooth optimization for the new problem, they get a solution technique for the initial problem. The efficiency of this approach is shown on a series of tough nonacademic problems from the area of optimum shape design and economic modelling.

Dealing with both, the theory and applications of optimization problems with equilibrium constraints, this book is extremely useful for researchers in optimization theory and for applied mathematicians.

Reviewer: W.W.Breckner (Cluj-Napoca)

### MSC:

90C26 | Nonconvex programming, global optimization |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |