Chapter 7: Spectrahedral Approximations of Convex Hulls of Algebraic Sets

This chapter describes a method for finding spectrahedral approximations of the convex hull of a real algebraic variety (the set of real solutions to a finite system of polynomial equations). The procedure creates a nested sequence of convex approximations of the convex hull of the variety. Computations can be done modulo the ideal generated by the polynomials which has several advantages. We examine conditions under which the sequence of approximations converges to the closure of the convex hull of the real variety, either asymptotically or in finitely many steps, with special attention to the case in which the very first approximation yields a semidefinite representation of the convex hull. These methods allow optimization, or approximation of the optimal value, of a linear function over a real algebraic variety via semidefinite programming.

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