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In this paper, we study an uncertain inequality system, where the input data are uncertain and belong to prescribed uncertainty sets. Using the deterministic approach in robust optimization, we treat this uncertain system by examining the so-called robust system. This approach enables us to compute the second order tangent sets for the solution set of the robust system and then obtain the second order epi-subderivative for the indicator function of its solution set. In this way, we are able to calculate the graphical derivative for the normal cone mapping of solution set of the robust system under certain qualification conditions. As applications, we establish second order necessary and sufficient optimality conditions, and derive necessary and sufficient conditions for stability properties such as the isolated calmness of optimization problems involving uncertain constraints under weak qualification conditions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a convex quadratic multiobjective optimization problem, where both the objective and constraint functions involve data uncertainty. We employ a deterministic approach to examine robust optimality conditions and find robust (weak) Pareto solutions of the underlying uncertain multiobjective problem. We first present new necessary and sufficient conditions in terms of linear matrix inequalities for robust (weak) Pareto optimality of the multiobjective optimization problem. We then show that the obtained optimality conditions can be alternatively checked via other verifiable criteria including a robust Karush–Kuhn–Tucker condition. Moreover, we establish that a (scalar) relaxation problem of a robust weighted-sum optimization program of the multiobjective problem can be solved by using a semidefinite programming (SDP) problem. This provides us with a way to numerically calculate a robust (weak) Pareto solution of the uncertain multiobjective problem as an SDP problem that can be implemented using, e.g., MATLAB. [ABSTRACT FROM AUTHOR]

In this paper we show that two-stage adjustable robust linear programs with affinely adjustable data in the face of box data uncertainties under separable quadratic decision rules admit exact semi-definite program (SDP) reformulations in the sense that they share the same optimal values and admit a one-to-one correspondence between the optimal solutions. This result allows adjustable robust solutions of these robust linear programs to be found by simply numerically solving their SDP reformulations. We achieve this result by first proving a special sum-of-squares representation of non-negativity of a separable non-convex quadratic function over box constraints. Our reformulation scheme is illustrated via numerical experiments by applying it to an inventory-production management problem with the demand uncertainty. They demonstrate that our separable quadratic decision rule method to two-stage decision-making performs better than the single-stage approach and is capable of solving the inventory production problem with a greater degree of uncertainty in the demand. [ABSTRACT FROM AUTHOR]

In this paper, we examine the properly twice epi-differentiability and compute the second order epi-subderivative of the indicator function to a class of sets including the finite union of parabolically derivable and parabolically regular sets. In this way, we provide no-gap second order optimality conditions for a disjunctive constrained problem. Moreover, we derive applications of our results to some types of disjunctive programs. [ABSTRACT FROM AUTHOR]

In this paper, we propose a bounded degree hierarchy of both primal and dual conic programming relaxations involving both semi-definite and second-order cone constraints for solving a nonconvex polynomial optimization problem with a bounded feasible set. This hierarchy makes use of some key aspects of the convergent linear programming relaxations of polynomial optimization problems (Lasserre in Moments, positive polynomials and their applications, World Scientific, Singapore, 2010) associated with Krivine–Stengle's certificate of positivity in real algebraic geometry and some advantages of the scaled diagonally dominant sum of squares (SDSOS) polynomials (Ahmadi and Hall in Math Oper Res, 2019. 10.1287/moor.2018.0962; Ahmadi and Majumdar in SIAM J Appl Algebra Geom 3:193–230, 2019). We show that the values of both primal and dual relaxations converge to the global optimal value of the original polynomial optimization problem under some technical assumptions. Our hierarchy, which extends the so-called bounded degree Lasserre hierarchy (Lasserre et al. in Eur J Comput Optim 5:87–117, 2017), has a useful feature that the size and the number of the semi-definite and second-order cone constraints of the relaxations are fixed and independent of the step or level of the approximation in the hierarchy. As a special case, we provide a convergent bounded degree second-order cone programming (SOCP) hierarchy for solving polynomial optimization problems. We then present finite convergence at step one of the SOCP hierarchy for classes of polynomial optimization problems. This includes one-step convergence for a new class of first-order SDSOS-convex polynomial programs. In this case, we also show how a global solution is recovered from the level one SOCP relaxation. We finally derive a corresponding convergent conic linear programming hierarchy for conic-convex semi-algebraic programs. Whenever the semi-algebraic set of the conic-convex program is described by concave polynomial inequalities, we show further that the values of the relaxation problems converge to the common value of the convex program and its Lagrangian dual under a constraint qualification. [ABSTRACT FROM AUTHOR]

In this paper, under the existence of a certificate of nonnegativity of the objective function over the given constraint set, we present saddle-point global optimality conditions and a generalized Lagrangian duality theorem for (not necessarily convex) polynomial optimization problems, where the Lagrange multipliers are polynomials. We show that the nonnegativity certificate together with the archimedean condition guarantees that the values of the Lasserre hierarchy of semidefinite programming (SDP) relaxations of the primal polynomial problem converge asymptotically to the common primal-dual value. We then show that the known regularity conditions that guarantee finite convergence of the Lasserre hierarchy also ensure that the nonnegativity certificate holds and the values of the SDP relaxations converge finitely to the common primal-dual value. Finally, we provide classes of nonconvex polynomial optimization problems for which the Slater condition guarantees the required nonnegativity certificate and the common primal-dual value with constant multipliers and the dual problems can be reformulated as semidefinite programs. These classes include some separable polynomial programs and quadratic optimization problems with quadratic constraints that admit certain hidden convexity. We also give several numerical examples that illustrate our results. [ABSTRACT FROM AUTHOR]

This study evaluated effects of differing gel volumes on pharmacokinetics (PK). IQB4012, a gel containing the non-nucleoside reverse transcriptase inhibitor IQP-0528 and tenofovir (TFV), was applied to the pigtailed macaque vagina and rectum. Vaginal gel volumes (1% loading of both drugs) were 0.5 or 1.5 ml; following wash-out, 1 or 4 ml of gel were then applied rectally. Blood, vaginal, and rectal fluids were collected at 0, 2, 4, and 24 h. Vaginal and rectal tissue biopsies were collected at 4 and 24 h. There were no statistically significant differences in concentrations for either drug between gel volumes within compartments at matched time points. After vaginal gel application, median IQP-0528 concentrations were ~ 104-105 ng/g, 105-106 ng/ml, and 103-105 ng/ml in vaginal tissues, vaginal fluids, and rectal fluids, respectively (over 24 h). Median vaginal TFV concentrations were 1-2 logs lower than IQP-0528 levels at matched time points. After rectal gel application, median IQP-0528 and TFV concentrations in rectal fluids were ~ 103-105 ng/ml and ~ 102-103 ng/ml, respectively. Concentrations of both drugs sampled in rectal tissues were low (~ 101-103 ng/g). For 1 ml gel, half of sampled rectal tissues had undetectable concentrations of either drug, and over half of sampled rectal fluids had undetectable TFV concentrations. These results indicate differences in drug delivery between the vaginal and rectal compartments, and that smaller vaginal gel volumes may not significantly compromise microbicide PK and prophylactic potential. However, effects of rectal gel volume on PK for both drugs were less definitive. [ABSTRACT FROM AUTHOR]

This paper studies a bilevel polynomial program involving box data uncertainties in both its linear constraint set and its lower-level optimization problem. We show that the robust global optimal value of the uncertain bilevel polynomial program is the limit of a sequence of values of Lasserre-type hierarchy of semidefinite linear programming relaxations. This is done by first transforming the uncertain bilevel polynomial program into a single-level non-convex polynomial program using a dual characterization of the solution of the lower-level program and then employing the powerful Putinar's Positivstellensatz of semi-algebraic geometry. We provide a numerical example to show how the robust global optimal value of the uncertain bilevel polynomial program can be calculated by solving a semidefinite programming problem using the MATLAB toolbox YALMIP. [ABSTRACT FROM AUTHOR]

We present an exact formula for the radius of robust feasibility of uncertain linear programs with a compact and convex uncertainty set. The radius of robust feasibility provides a value for the maximal 'size' of an uncertainty set under which robust feasibility of the uncertain linear program can be guaranteed. By considering spectrahedral uncertainty sets, we obtain numerically tractable radius formulas for commonly used uncertainty sets of robust optimization, such as ellipsoids, balls, polytopes and boxes. In these cases, we show that the radius of robust feasibility can be found by solving a linearly constrained convex quadratic program or a minimax linear program. The results are illustrated by calculating the radius of robust feasibility of uncertain linear programs for several different uncertainty sets. [ABSTRACT FROM AUTHOR]

In this paper we study two-stage affinely adjustable robust multi-objective optimization problems. We show how (weak) Pareto optimal solutions of these robust multi-objective problems can be found by solving conic linear programming problems. We do this by first deriving numerically verifiable conditions that characterize (weak) Pareto optimal solutions of affinely adjustable robust multi-objective programs under a spectrahedron uncertainty set. The uncertainty set covers most of the commonly used uncertainty sets of robust optimization. We then reformulate the weighted-sum optimization problems of the multi-objective problems, derived with the aid of the optimality conditions, as equivalent conic linear programming problems, such as semidefinite programs or second-order cone programs, to find the (weak) Pareto optimal solutions. We illustrate by an example how our results can be used to find a second-stage (weak) Pareto optimal solution by solving a semidefinite program using a commonly available software. [ABSTRACT FROM AUTHOR]

The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained. [ABSTRACT FROM AUTHOR]

The Fréchet and limiting second-order subdifferentials of a proper lower semicontinuous convex function $\varphi: \mathbb R^n\rightarrow\bar{\mathbb R}$ have a property called the positive semi-definiteness (PSD)-in analogy with the notion of positive semi-definiteness of symmetric real matrices. In general, the PSD is insufficient for ensuring the convexity of an arbitrary lower semicontinuous function φ. However, if φ is a C function then the PSD property of one of the second-order subdifferentials is a complete characterization of the convexity of φ. The same assertion is valid for C functions of one variable. The limiting second-order subdifferential can recognize the convexity/nonconvexity of piecewise linear functions and of separable piecewise C functions, while its Fréchet counterpart cannot. [ABSTRACT FROM AUTHOR]

This paper deals with the generalized Clarke epiderivative of the extremum (or efficient point) multifunction in parametric vector optimization problems. The formulas for computing and/or estimating the generalized Clarke epiderivative of this extremum multifunction are given in terms of the Clarke tangent cone to the graph of a multifunction or the constraint mapping and/or the Fréchet derivative of the objective function. An application to semi-infinite programming is given. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of continuity properties of Pareto solution maps for parametric semi-infinite vector optimization problems (PSVO). We establish new necessary conditions for lower and upper semicontinuity of Pareto solution maps under functional perturbations of both objective functions and constraint sets. We also show that the necessary condition becomes sufficient for the lower and upper semicontinuous properties in the special case where the constraint set mapping is lower semicontinuous at the reference point. Examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]