# zbMATH — the first resource for mathematics

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.
Full Text: