Department of Computer Science | Institute of Theoretical Computer Science | CADMO
Theory of Combinatorial Algorithms
Prof. Emo Welzl and Prof. Bernd Gärtner
Department of Computer Science | Institute of Theoretical Computer Science | CADMO
Prof. Emo Welzl and Prof. Bernd Gärtner
| Mittagsseminar Talk Information |
Date and Time: Tuesday, May 12, 2026, 12:15 pm
Duration: 30 minutes
Location: OAT S15
Speaker: Manuel Wiedmer
Efficient algorithms for convex optimization, such as the ellipsoid method, require an a priori bound on the radius of a ball around the origin that contains 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. 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, simultaneously, Khachiyan-type examples do not apply.
In this talk, I will discuss our recent result about programs minimizing a convex polynomial (of arbitrary degree) over a polyhedron. We prove that if an optimal solution exists, then there is an approximately optimal one with polynomial bit length as well. Combined with the ellipsoid method, this yields the first polynomial-time algorithm for (approximate) convex polynomial programming. Our result relies on a structural decomposition of any convex polynomial into a sum of a linear function and a polynomial on a linear subspace that admits a strongly convex lower bound, where the logarithm of the strong convexity parameter is at most polynomial in the input size.
This talk is based on joint work with Lucas Slot and David Steurer.
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