Optimization Problems over Noncompact Semialgebraic Sets

Abstract

In this talk, we will introduce some recent progress in dealing with optimization problems over noncompact semialgebraic sets. We will start with the problem of optimizing a parametric linear function over a noncompact real algebraic variety. Then we will introduce how to compute the semidefinite representation or approximation of the convex hull of a noncompact semialgebraic set. Finally, we will show how to characterize the lifts of noncompact convex sets by the cone factorizations of properly defined slack operators.Optimizing a parametric linear function over a real algebraic variety We consider the problem of optimizing a parametric linear function over a real algebraic variety. . , Xn] and c = (c1, . . . , cn) denotes unspecified parameters. The optimal value c * 0 can be regarded as a function of the parameters c, i.e. the optimal value function.Our goal is to compute a polynomial Φ ∈ Q[c0, c1, . . . , cn] that defines a hypersurface in the parameters' space which contains the graph of this function.Assuming that V is irreducible, smooth and compact in R n , Rostalski and Sturmfels showed that the optimal value function is represented by the defining equation of the irreducible hypersurface V * dual to the projective closure of V in R n [11, Theorem 5.23]. Let C = cl (co (V ∩ R n )) be the closure of the convex hull of V ∩ R n , in [5, Theorem 1.5], we proved that this conclusion is still true for a noncompact real algebraic variety V ∩ R n when it is irreducible, smooth and the recession cone 0 + C of C is pointed (contains no lines). Although the conclusion is correct, the proof of [5, Theorem 1.5] works only if the convex hull of V ∩ R n is closed. The new proof can be found in [4, Corollary 3.2]. Moreover, in [4], we showed (−c * 0 : γ1 : · · · : γn) ∈ V * whenever the optimal value c * 0 is bounded at (γ1, . . . , γn) and V is smooth. When V is not smooth but its real trace is compact, we constructed recursively a finite number of dual varieties such that (−c * 0 : γ1 : · · · : γn) lies in the union of these dual varieties. For some special parameters' values, the representing polynomials of the dual variety V * can be identically zero, which give no information on the optimal value. We designed a parametric variant of [3] that identifies those regions of the parameters' space and computed for each of these regions a new polynomial defining the optimal value over the considered region.