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SQP-methods for semiinfinite optimization problems. (SQP-Methoden für semiinfinite Optimierungsprobleme.) (German) Zbl 0743.90104

Trier: Univ. Trier, FB IV, 101 S. (1990).
This doctoral dissertation is concerned with the following problem of semi-infinite programming: Minimize \(f(x)\) on \(X\), where \(X=\{x\in\mathbb{R}^ n\mid g(x,y)\geq0\) for all \(y\in Y\}\) and \(Y=\{y\in\mathbb{R}^ m\mid h_ s(y)\geq0\), \(s\in S\}\), under the following assumptions: \(S\) a finite set, \(f\in C^ \alpha(\mathbb{R}^ n,\mathbb{R})\), \(g\in C^ \alpha(\mathbb{R}^ n\times\mathbb{R}^ m,\mathbb{R})\), \(\alpha\geq0\) and \(h_ s\in C^ 2(\mathbb{R}^ m,\mathbb{R})\) for every \(s\in S\). It is further assumed that \(Y\) is compact and regular, i.e., for every \(y\in Y\) the linear independence constraint qualification is satisfied.
For the solution of this problem at first a conceptual algorithm (see Chapter 3) is considered which consists of two steps: (1) Starting with some \(x_ 0\in\mathbb{R}^ n\) determine, for \(k=0,1,2,\ldots,\) all the local minima of \(g(x_ k,y)\) on \(Y\). Assume that these are finitely many, denoted by \(y^ 1(x_ k),\ldots,y^ r(x_ k)\). (2) Minimize \(f(x)\) subject to \(g(x,y^ i(x))\geq0\) \((i=1,\ldots,r)\), where \(y^ i(x)\) is a local minimizer of \(g(x,y)\) on \(Y\).
The second step is based on the following theorem (see Theorem 1.3):
Let \(\bar y\in Y\) be a non-degenerate local minimizer of \(g(\bar x,y)\) on \(Y\). Then there is a neighbourhood \(U(\bar x)\) of \(\bar x\) and a continuously differentiable function \(y:U(\bar x)\to\mathbb{R}^ m\) such that \(y(\bar x)=\bar y\) and for all \(x\in U(\bar x)\) we have (a) \(y(x)\in Y\) and \(S_ 0(y(x))=S_ 0(\bar y)\) where, for every \(y\in Y\), \(S_ 0(y)=\{s\in S\mid h_ s(y)=0\}\) and (b) \(y(x)\) is a non-degenerate local minimizer of \(g(x,y)\) on \(Y\).
In Chapter 4 local convergence and in Chapter 5 global convergence to local minimizers of \(f\) on \(X\) are investigated. Chapter 6 is devoted to an implementation of the conceptual algorithm described above which is based on a connection of the concepts of recursive quadratic programming, quasi-Newton-methods (BFGS), globalization with \(\ell_ 1\)-penalty functions as “merit functions”, and local reduction. In Chapter 7 the implemented algorithm is demonstrated by several test problems, the last being a problem of robot control whose numerical results, however, are not given but rather referred to in corresponding papers by the author, Haaren, Hettich and Kortanek.

MSC:

90C34 Semi-infinite programming
90C31 Sensitivity, stability, parametric optimization
49M30 Other numerical methods in calculus of variations (MSC2010)
90-08 Computational methods for problems pertaining to operations research and mathematical programming
90C20 Quadratic programming
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