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This paper is devoted to second-order optimality conditions of constrained vector optimization problems with continuously Fréchet differentiable data. Different from conventional approaches, we employ a new one, image space analysis, to establish suitable regularity conditions and then present new second-order necessary optimality conditions that depend on one single KKT multiplier for all critical directions. Moreover, we propose one more regularity condition for the multiobjective case and compare it with the previous one. Besides, we compare the main regularity conditions with the existing ones in the literature for the scalar case. [ABSTRACT FROM AUTHOR]

This paper presents exact dual semi-definite programs (SDPs) for robust SOS-convex polynomial optimization problems with affinely adjustable variables in the sense that the optimal values of the robust problem and its associated dual SDP are equal with the solution attainment of the dual problem. This class of robust convex optimization problems includes the corresponding quadratically constrained convex quadratic optimization problems and separable convex polynomial optimization problems, and it employs a general bounded spectrahedron uncertainty set that covers the most commonly used uncertainty sets of numerically solvable robust optimization models, such as boxes, balls and ellipsoids. As special cases, it also demonstrates that explicit exact dual SDP and second-order cone programming (SOCP) in terms of original data hold for the robust two-stage convex quadratic programs with quadratic constraints and the robust two-stage separable convex quadratic programs under an ellipsoidal uncertainty set, respectively. Finally, the paper illustrates the results via numerical implementations of the developed SDP duality scheme on adjustable robust lot-sizing problems with nonlinear costs under demand uncertainty. [ABSTRACT FROM AUTHOR]

In this paper, we examine a zero-duality gap with polynomial multipliers between a non-convex polynomial program and its associated generalized Lagrangian dual problem. To this end, we introduce a sum of squares certificate of non-negativity, abbreviated as (SC), that allows us to verify the non-negativity of a polynomial over a feasible set of a system of given polynomials. With the help of (SC), we show that the zero-duality gap with polynomial multipliers coincides with the convergent Lasserre hierarchy of semidefinite programming relaxations. This zero-duality gap with polynomial multipliers reduces to the classical zero-duality gap with constant multipliers when considering it in the setting of SOS-convex polynomial programs or quadratic programs with the hidden convexity. We also show that, in the framework of truncated polynomial systems, the zero-duality gap with polynomial multipliers can be checked via the classical Karush–Kuhn–Tucker condition under an additional convexity condition. [ABSTRACT FROM AUTHOR]

The radius of robust feasibility is the maximal size of uncertain set in which the robust feasible set for an uncertain program is nonempty. In this paper, we employ robust optimization technique to study a class of uncertain convex conic program, and give its formulas for radius of robust feasibility under several data uncertain sets. First, with aid of the distance from the origin to the so-called epigraphcal set, we provide computable upper and lower bounds of the radius of robust feasibility for convex conic program in face of ball uncertainty. Second, a formula is presented for the radius of robust feasibility for robust convex optimization problem with SOS-convex polynomial constraints under ball uncertain sets. Finally, some exact formulas of radius of robust feasibility are given for convex conic program with piecewise linear function constraints under boxes or polytopes uncertain sets. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a multiobjectif bilevel programming problem with set valued constraints in both the upper problem and the lower problem. Using the notion of the support function to the feasible set mapping, together with an appropriate constraint qualification, we establish necessary optimality conditions in terms of asymptotical pointwise compact approximations; continuity of data functions and compactness of approximations are not necessarily required. In order to get sufficient optimality conditions, we introduce generalized pseudo-invexity and quasi-invexity notions in terms of asymptotical pointwise compact approximations. Examples that illustrate our results are given. [ABSTRACT FROM AUTHOR]

This paper mainly focuses on optimality conditions for efficient solutions of a nonconvex constrained multiobjective optimization problem via image space analysis. By virtue of the Gerstewitz function and the sum of the components of the objective function, a nonlinear regular weak separation function for efficient solutions, is constructed. In order to illustrate the importance of the sum of the components of the objective function in the nonlinear regular weak separation function for efficient solutions, a nonlinear regular weak separation function for weak efficient solutions, is also given. Moreover, a nonlinear strong separation function for efficient and weak efficient solutions, is introduced. Then, some theorems of the weak and strong alternative and a global necessary and sufficient optimality condition for efficient solutions are derived by means of the nonlinear regular weak and strong separation functions for efficient solutions. A saddle point sufficient optimality condition for efficient solutions in terms of a generalized Lagrangian function associated with the nonlinear regular weak separation function, is established. Finally, under some suitable assumptions, a saddle point necessary optimality condition is obtained, which further yields a Karush/Kuhn–Tucker necessary optimality condition for efficient solutions by the Clarke subdifferential. [ABSTRACT FROM AUTHOR]

Augmented Lagrangian methods are effective tools for solving large-scale nonlinear programming problems. At each outer iteration, a minimization subproblem with simple constraints, whose objective function depends on updated Lagrange multipliers and penalty parameters, is approximately solved. When the penalty parameter becomes very large, solving the subproblem becomes difficult; therefore, the effectiveness of this approach is associated with the boundedness of the penalty parameters. In this paper, it is proved that under more natural assumptions than the ones employed until now, penalty parameters are bounded. For proving the new boundedness result, the original algorithm has been slightly modified. Numerical consequences of the modifications are discussed and computational experiments are presented. [ABSTRACT FROM AUTHOR]