Equivalence of saddle-points and optima, and duality for a class of non- smooth non-convex problems. (English) Zbl 0642.49018

The author considers the problem of minimizing a locally Lipschitz function \(f: {\mathbb{R}}^ n\to {\mathbb{R}}\) subject to the constraints \(g_ i(x)\leq 0\), \(i=1,...,m\), where each \(g_ i:{\mathbb{R}}^ n\to {\mathbb{R}}\) is locally Lipschitz. After forming the Lagrangian \[ L(x;\lambda):=f(x)+\sum^{m}_{i=1}\lambda_ ig_ i(x), \] he uses invexity hypotheses on f, \(g_ i\), and a constraint qualification to prove that a point \(\hat x\) provides a global minimum for this problem iff there is a corresponding \({\hat \lambda}\in {\mathbb{R}}^ m_+\) satisfying the saddle-point condition \(L(\hat x;\lambda)\leq L(\hat x;{\hat \lambda})\leq L(x;{\hat \lambda})\), \(x\in {\mathbb{R}}^ n\), \(\lambda \in {\mathbb{R}}^ m_+\). The relation of this condition to the generalized Kuhn-Tucker and Fritz John optimality conditions is discussed, as are certain forms of duality. (A locally Lipschitz function \(f: {\mathbb{R}}^ n\to {\mathbb{R}}\) is “invex” if there exists a function \(\eta: {\mathbb{R}}^ n \times {\mathbb{R}}^ n \to {\mathbb{R}}^ n\) for which \(f(y)-f(x) \geq \max \{<\xi,\eta (x,y)>:\zeta\in \partial f(x)\}\), where \(\partial f(x)\) is Clarke’s generalized gradient of f at x.)
Reviewer: P.Loewen


49K10 Optimality conditions for free problems in two or more independent variables
49N15 Duality theory (optimization)
90C30 Nonlinear programming
26B35 Special properties of functions of several variables, Hölder conditions, etc.
90C25 Convex programming
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