Hesse's Redemption: Efficient Convex Polynomial Programming

2025-11-05

AI Summary

This paper resolves two fundamental open problems in convex polynomial programming: (1) existence and boundedness characterization of global optima for unconstrained convex polynomials of degree four or higher; and (2) existence of polynomial-bit-length approximate optimal solutions for convex polynomials of arbitrary degree under polyhedral constraints. Moving beyond conventional approaches reliant on linear optimality characterizations (e.g., KKT conditions), the authors develop a novel, KKT-free framework for analyzing solution boundedness. They establish, for the first time, that ε-approximate optimal solutions admit polynomial bit complexity—positively resolving an open question posed by Nesterov. Integrating the ellipsoid method with a new boundary estimation technique, they design the first polynomial-time approximation algorithm applicable to convex polynomials of arbitrary degree, encompassing both unconstrained and polyhedrally constrained settings. This work bridges a long-standing theoretical and algorithmic gap between linear/convex quadratic programming and semidefinite programming.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Develop solution bounds for convex polynomial optimization without linear characterizations
• Establish polynomial bit length existence for approximately optimal solutions
• Provide first polynomial-time algorithm for convex polynomial programming

Innovation

• Developed solution bounds for convex polynomial programming
• Proved existence of polynomial-bit approximate optimal solutions
• Enabled polynomial-time algorithm via ellipsoid method