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Generalized Hessian matrix and second-order optimality conditions for problems with $$C^{1,1}$$ data. (English) Zbl 0542.49011
This paper has two objectives: 1. Use Clarke’s generalized derivative concept to generalize the ”Hessian” concept to C(1,1) functions; that is, to functions with locally Lipschitz gradients. For example, if f is $$C^ 2$$, then $$\{\max(f,0)\}^ 2$$ is C(1,1). 2. Use the properties of generalized Hessian including Taylor expansion, to derive second order optimality conditions for mathematical programming problems with nonlinear constraints and C(1,1) data.
The following two theorems are of interest: 1. Let $${\mathcal O}$$ be a nonempty open subset of $$R^ n$$. Let $$f\in C(1,1)({\mathcal O})$$ and [a,b]$$\subset {\mathcal O}$$. Then, there is $$c\in(a,b)$$ and $$M_ c\in \partial^ 2f(c)$$ such that $$f(b)=f(a)+<\nabla f(a),b-a>+{1\over2}<M_ c(b-a),b-a>.$$ Here, $$\partial^ 2f(c)$$ denotes the convex hull of the set of all limits of the form lim $$\nabla^ 2f(x_ i)$$, for all possible sequences $$\{x_ i\}$$ converging to c and for which f is twice differentiable and $$\nabla^ 2f(x_ i)$$ is meaningful.
2. Let $$x_ 0$$ be a local minimum of the constrained problem (c): Minimize f(x) subject to $$g_ i(x)\leq 0$$, $$i=1,...,m$$ and $$h_ j(x)=0$$, $$j=1,...,n$$; where f, $$g_ i$$, and $$h_ j$$ are all C(1,1). Let $$G(\lambda)=\{x| g_ i(x)=0$$ if $$\lambda_ i>0$$, $$g_ i(x)\leq 0$$ if $$\lambda_ i=0$$ and $$h_ j=0$$ for all $$j\}$$ and let $$T_{\lambda}$$ be the tangent cone to $$G(\lambda)$$ at $$x_ 0$$. Let L denote the Lagrangian $$L(x,\lambda,u)=f(x)+\sum \lambda_ ig_ i(x)+\sum \mu_ jh_ j(x)$$ and let $$\partial^ 2_{xx}L(x_ 0,\lambda,\mu)$$ denote the generalized Hessian of L at $$x_ 0$$. Then, for each Kuhn-Tucker multiplier ($$\lambda$$,$$\mu)$$ and for each $$d\in T_{\lambda}$$, there is a matrix $$A\in \partial^ 2_{xx}L(x_ 0,\lambda,\mu)$$ such that $$<Ad,d>\geq 0$$.
Reviewer: M.Sury

##### MSC:
 49K10 Optimality conditions for free problems in two or more independent variables 58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds 90C30 Nonlinear programming 49M37 Numerical methods based on nonlinear programming 26B05 Continuity and differentiation questions
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