Hesse's Redemption: Efficient Convex Polynomial Programming
Lucas Slot, David Steurer, Manuel Wiedmer
TL;DR
This work resolves the long-standing challenge of efficiently solving convex polynomial programs, including unconstrained minimization of convex polynomials of degree four and higher, by proving a structure theorem that decomposes any convex polynomial into a linear part plus a reduced-variable component admitting a strongly convex quadratic lower bound. The key technical insight is the Hessian determinant dichotomy: if the Hessian is not identically zero, one obtains a global $\mu$-strongly convex lower bound with $\mu$ bounded below by a single-exponential function of encoding length; if the Hessian vanishes everywhere, there exists a global direction of linearity, enabling a dimension-reducing reduction. This structure yields a singly-exponential bound on the norm of a global minimizer on a polyhedron, and when combined with the ellipsoid method, provides the first polynomial-time algorithm for convex polynomial programming of degree $\ge 4$, answering a question posed by Nesterov. The paper also surveys the complexity landscape, showing that while general polynomial optimization can lack compact witnesses, convexity restores exponentially bounded minimizers and permits polynomial-time approximate minimization; it discusses implications for SDP, SOS relaxations, and higher-order Newton methods, and clarifies the practical significance for algorithmic convex optimization and complexity theory.
Abstract
Efficient algorithms for convex optimization, such as the ellipsoid method, require an a priori bound on the radius of a ball around the origin guaranteed to contain an optimal solution if one exists. For linear and convex quadratic programming, such solution bounds follow from classical characterizations of optimal solutions by systems of linear equations. For other programs, e.g., semidefinite ones, examples due to Khachiyan show that optimal solutions may require huge coefficients with an exponential number of bits, even if we allow approximations. Correspondingly, semidefinite programming is not even known to be in NP. The unconstrained minimization of convex polynomials of degree four and higher has remained a fundamental open problem between these two extremes: its optimal solutions do not admit a linear characterization and, at the same time, Khachiyan-type examples do not apply. We resolve this problem by developing new techniques to prove solution bounds when no linear characterizations are available. Even for programs minimizing a convex polynomial (of arbitrary degree) over a polyhedron, we prove that the existence of an optimal solution implies that an approximately optimal one with polynomial bit length also exists. These solution bounds, combined with the ellipsoid method, yield the first polynomial-time algorithm for convex polynomial programming, settling a question posed by Nesterov (Math. Program., 2019). Before, no polynomial-time algorithm was known even for unconstrained minimization of a convex polynomial of degree four.