Quasiconvex sum-of-squares programming

A sum-of-squares program is an optimization problem with polynomial sum-of-squares constraints. The constraints and the objective function are affine in the decision variables. This paper introduces a generalized sum-of-squares programming problem. This generalization allows one decision variable to enter bilinearly in the constraints. The bilinear decision variable enters the constraints in a particular structured way. The objective function is the single bilinear decision variable. It is proved that this formulation is quasiconvex and hence the global optima can be computed via bisection. Many nonlinear analysis problems can be posed within this framework and two examples are provided.