多项式Optimization的代数次数
Algebraic Degree of Polynomial Optimization
摘要：
考虑目标和约束均由多元多项式描述的多项式最优化问题。在某些泛生性假设下，证明了优化器的最优性条件始终成立，且优化器的坐标是输入多项式系数的代数函数.我们还给出了最优坐标代数次数的一般公式。代数次数的推导等价于计算所有复临界点的个数。作为特例，我们得到了二次约束二次规划（QCQP）、二阶锥规划（SOCP）和$p$ -阶锥规划（pOCP）的代数度，类似于半定规划的代数度.
英文摘要：
Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials. We also give a general formula for the algebraic degree of the optimal coordinates. The derivation of the algebraic degree is equivalent to counting the number of all complex critical points. As special cases, we obtain the algebraic degrees of quadratically constrained quadratic programming (QCQP), second order cone programming (SOCP) and $p$-th order cone programming (pOCP), in analogy to the algebraic degree of semidefinite programming.
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