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Quadratic forms which are positive on a cone and quadratic duality. (Russian. English summary) Zbl 0552.49013
As has been suggested by J. J. Moreau [Fonctionnelles convexes. Séminaire sur les équations aux dérivées partielles, Collège de France, 1966(67)] the bilinear functional which occurs in polarity theory can be replaced by a nonlinear functional without invalidating many of the basic properties. In this paper, the author considers the homogeneous polarity generated by the functional \(\phi\) : SM\({}_ n\times R^ n\to R\) defined by \(\phi (A,x)=<Ax,x>,\) where \(SM_ n\) is the linear space of all real symmetric matrices of order n and \(<\cdot,\cdot >\) is the inner product in \(R^ n\); bipolar theorems with respect to \(SM_ n\) and \(R^ n\), respectively are established. As applications, the S-procedure [see A. L. Fradkov and V. A. Jakubovič, Vestn. Leningr. Univ. 1973, no. 1, 81-87 (1973; Zbl 0259.93033)] in control theory and Pareto optimization is considered. In particular, an interesting property of the positive cone in Euclidean spaces is obtained: For \(n\geq 5\), the positive cone \(R^ n_+\) is not quadratic finite generated, i.e. there does not exist a finite number of forms \(F_ 1,F_ 2,...,F_ N\) such that every positive form on \(R^ n_+\) is the sum of a positive definite form on \(R^ n\) and a nonnegative linear combination of \(F_ 1,F_ 2,...,F_ N.\)
Reviewer: T.Precupanu

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
15A63 Quadratic and bilinear forms, inner products
49N15 Duality theory (optimization)
15B48 Positive matrices and their generalizations; cones of matrices
90C20 Quadratic programming
58E17 Multiobjective variational problems, Pareto optimality, applications to economics, etc.
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