Abstract In this paper, we analyze conic minimax convex polynomial optimization problems. Under a suitable regularity condition, an exact conic programming relaxation is established based on a positivity characterization of a max function over a conic convex system. Further, we consider a general conic minimax $$\rho $$ ρ -convex polynomial optimization problem, which is defined by appropriately extending the notion of conic convexity of a vector-valued mapping. For this problem, it is shown that a Karush-Kuhn-Tucker condition at a global minimizer is necessary and sufficient for ensuring an exact relaxation with attainment of the conic programming relaxation. The exact conic programming relaxations are applied to SOS-convex polynomial programs, where appropriate choices of the data allow the associated conic programming relaxation to be reformulated as a semidefinite programming problem. In this way, we can further elaborate the obtained results for other special settings including conic robust SOS-convex polynomial problems and difference of SOS-convex polynomial programs.

Abstract This paper presents a robust framework for handling a conic multiobjective linear optimization problem, where the objective and constraint functions are involving affinely parameterized data uncertainties. More precisely, we examine optimality conditions and calculate efficient solutions of the conic robust multiobjective linear problem. We provide necessary and sufficient linear conic criteria for efficiency of the underlying conic robust multiobjective linear program. It is shown that such optimality conditions can be expressed in terms of linear matrix inequalities and second-order conic conditions for a multiobjective semidefinite program and a multiobjective second order conic program, respectively. We show how efficient solutions of the conic robust multiobjective linear problem can be found via its conic programming reformulation problems including semidefinite programming and second-order cone programming problems. Numerical examples are also provided to illustrate that the proposed conic programming reformulation schemes can be employed to find efficient solutions for concrete problems including those arisen from practical applications.

Abstract The main purpose of this paper is to investigate the existence of global optimal solutions for nonsmooth and nonconvex robust optimization problems. To do this, we first introduce a concept called extended tangency variety and show how a robust optimization problem can be transformed into a minimizing problem of the corresponding tangency variety. We utilize this concept together with a constraint qualification condition and the boundedness of the objective function to provide relationships among the concepts of robust properness, robust M-tamesness and robust Palais-Smale condition related to the considered problem. The obtained results are also employed to derive necessary and sufficient conditions for the existence of global optimal solutions to the underlying robust optimization problem.

Abstract In this paper, we consider a polynomial problem with equilibrium constraints in which the constraint functions and the equilibrium constraints involve data uncertainties. Employing a robust optimization approach, we examine the uncertain equilibrium constrained polynomial optimization problem by establishing lower bound approximations and asymptotic convergences of bounded degree diagonally dominant sum-of-squares (DSOS), scaled diagonally dominant sum-of-squares (SDSOS) and sum-of-squares (SOS) polynomial relaxations for the robust equilibrium constrained polynomial optimization problem. We also provide numerical examples to illustrate how the optimal value of a robust equilibrium constrained problem can be calculated by solving associated relaxation problems. Furthermore, an application to electric vehicle charging scheduling problems under uncertain discharging supplies shows that for the lower relaxation degrees, the DSOS, SDSOS and SOS relaxations obtain reasonable charging costs and for the higher relaxation degrees, the SDSOS relaxation scheme has the best performance, making it desirable for practical applications.