Hesse的救赎：高效的凸多项式规划
Hesse's Redemption: Efficient Convex Polynomial Programming
AI总结：
针对四次及以上凸多项式无约束最小化这一开放问题，本文提出在无线性刻画下证明解界限的新技术，证明了多面体上凸多项式规划存在多项式比特长度的近似最优解，结合椭球法首次给出了凸多项式规划的多项式时间算法。
摘要：
凸优化的高效算法，如椭球法，需要预先设定一个以原点为中心的球半径界限，以保证在最优解存在时该球内必定包含最优解。对于线性和凸二次规划，这种解的界限可由线性方程组对最优解的经典刻画推导得出。对于其他规划（例如半定规划），Khachiyan给出的例子表明，即使允许近似，最优解也可能需要具有指数级比特数的巨大系数。相应地，半定规划甚至不知是否属于NP。四次及更高次凸多项式的无约束最小化一直处于这两种极端之间，是一个基本的开放问题：其最优解不满足线性刻画，同时Khachiyan型例子也不适用。我们通过开发在无线性刻画时证明解界限的新技术来解决此问题。即使对于在多面体上最小化凸多项式（任意次数）的规划，我们证明了最优解的存在意味着也存在具有多项式比特长度的近似最优解。这些解界限结合椭球法，产生了凸多项式规划的首个多项式时间算法，解决了Nesterov（Math. Program., 2019）提出的问题。此前，即使是四次凸多项式的无约束最小化，也不存在已知的多项式时间算法。
英文摘要：
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.
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