Algebraic Degree of Polynomial Optimization

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, 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 [J. Nie, K. Ranestad, and B. Sturmfels, The algebraic degree of semidefinite programming, Math. Programm., to appear].

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