Spectral Approach to Duality in Nonconvex Global Optimization

Abstract

A nonconvex problem of constrained optimization is analyzed in terms of its ordinary Lagrangian function. New sufficient conditions are obtained for the duality gap to vanish. Among them, the main condition is that the objective and constraint functions be the sums of convex functionals and nonconvex quadratic forms with certain specific spectral properties. The proofs are related to extensions of the classic Toeplitz--Hausdorff theorem, which states that a continuous quadratic mapping $\left( y_1, y_2 \right) = \left[ {\cal B}_1(z), {\cal B}_2(z) \right]$ from a complex Hilbert space $H = \{ z \}$ into $\footnotesize{\Bbb R}^2 = \left\{ \left( y_1,y_2 \right) \right\}$ transforms the unit sphere $|z| =1$ into a convex set. The extensions deal with a quadratic mapping $\left[ {\cal B}_1(z), \ldots, {\cal B}_k(z) \right]$ from a real Hilbert space into $\footnotesize{\Bbb R}^k$ with k being arbitrary. Applications to linear-quadratic optimal control theory are considered.