Face Of Data Uncertainty Research Articles

Robust error bounds for uncertain convex inequality systems with applications

In this paper, we devote ourselves to studying robust error bounds for a convex inequality system in the face of data uncertainty. Under the assumption that uncertain sets are convex compact, we present a sufficient condition for the existence of robust error bounds of the uncertain convex inequality system by means of the associated recession cone and recession functions. When the uncertain sets are only compact, we establish a necessary and sufficient condition for the uncertain convex inequality system to possess a robust error bound by the Ekeland variational principle. As an immediate application, we obtain robust error bounds for the uncertain convex polynomial inequality system.

Robust penalty function method for an uncertain multi-time control optimization problems

This paper gives some new results on multi-time first-order PDE constrained control optimization problem in the face of data uncertainty (MCOPU). We obtain the robust sufficient optimality conditions for (MCOPU). Further, we construct an unconstrained multi-time control optimization problem (MCOPU)ϱ corresponding to (MCOPU) via absolute value penalty function method. Then, we show that the robust optimal solution to the constrained problem and a robust minimizer to the unconstrained problem are equivalent under suitable hypotheses. Moreover, we give some non-trivial examples to validate the results established in this paper.

Parametric approach for approximate efficiency of robust multiobjective fractional programming problems

In this paper, a methodology is developed to solve an uncertain multiobjective fractional programming problem in the face of data uncertainty in the objective and constraint functions. We use the robust optimization approach (worst‐case approach) for finding ɛ‐efficient solutions of the associated robust multiobjective fractional programming problem defined as a robust (worst‐case) counterpart. For the robust multiobjective fractional programming problem constructed in the robust approach, In fact, in the parametric Dinkelbach approach. We define for the given multiobjective fractional programming problem its corresponding parametric robust vector optimization problem. Then, we establish both necessary and sufficient optimality conditions for a feasible solution to be an ‐efficient solution (an approximate efficient solution) of the parametric robust vector optimization problem. Also we prove the relationship between ɛ‐efficiency of the robust multiobjective fractional programming problem (and thus ɛ‐efficiency of the considered uncertain multiobjective fractional programming problem) and ‐efficiency of its corresponding parametric robust vector optimization problem constructed in the Dinkelbach approach. In order to prove this result, we also use a scalarization method. Thus, the equivalence between an approximate efficient solution of the parametric robust vector optimization problem and an approximate minimizer of its associated scalar optimization problem constructed in the used scalarization method is also established.

Classifying ecosystem stressor interactions: Theory highlights the data limitations of the additive null model and the difficulty in revealing ecological surprises.

Understanding how multiple co-occurring environmental stressors combine to affect biodiversity and ecosystem services is an on-going grand challenge for ecology. Currently, progress has been made through accumulating large numbers of smaller-scale empirical studies that are then investigated by meta-analyses to detect general patterns. There is particular interest in detecting, understanding and predicting 'ecological surprises' where stressors interact in a non-additive (e.g. antagonistic or synergistic) manner, but so far few general results have emerged. However, the ability of the statistical tools to recover non-additive interactions in the face of data uncertainty is unstudied, so crucially, we do not know how well the empirical results reflect the true stressor interactions. Here, we investigate the performance of the commonly implemented additive null model. A meta-analysis of a large (545 interactions) empirical dataset for the effects of pairs of stressors on freshwater communities reveals additive interactions dominate individual studies, whereas pooling the data leads to an antagonistic summary interaction class. However, analyses of simulated data from food chain models, where the underlying interactions are known, suggest both sets of results may be due to observation error within the data. Specifically, we show that the additive null model is highly sensitive to observation error, with non-additive interactions being reliably detected at only unrealistically low levels of data uncertainty. Similarly, plausible levels of observation error lead to meta-analyses reporting antagonistic summary interaction classifications even when synergies co-dominate. Therefore, while our empirical results broadly agree with those of previous freshwater meta-analyses, we conclude these patterns may be driven by statistical sampling rather than any ecological mechanisms. Further investigation of candidate null models used to define stressor-pair interactions is essential, and once any artefacts are accounted for, the so-called 'ecological surprises' may be more frequent than was previously assumed.

Robust constrained best approximation with nonconvex constraints

In this paper, we consider the set D of inequalities with nonconvex constraint functions in the face of data uncertainty. We show under a suitable condition that “perturbation property” of the robust best approximation to any $$x\in {\mathbb {R}}^n$$ from the set $$\tilde{K}:={{\bar{C}}} \cap D$$ is characterized by the strong conical hull intersection property (strong CHIP) of $${\bar{C}}$$ and D. The set C is an open convex subset of $${\mathbb {R}}^n$$ and the set D is represented by $$D:=\{x\in {\mathbb {R}}^n: g_{j}(x,v_j)\le 0, \; \forall \; v_j\in V_j, \; j=1,2,\ldots ,m\},$$ where the functions $$g_j:{\mathbb {R}}^n\times V_j\longrightarrow {\mathbb {R}}, \; j=1,2,\ldots ,m,$$ are continuously Frechet differentiable that are not necessarily convex, and $$v_j$$ is the uncertain parameter which belongs to an uncertainty set $$V_j\subset {\mathbb {R}}^{q_j}, \; j=1,2,\ldots ,m.$$ This is done by first proving a dual cone characterization of the robust constraint set D. Finally, following the robust optimization approach, we establish Lagrange multiplier characterizations of the robust constrained best approximation that is immunized against data uncertainty under the robust nondegeneracy constraint qualification. Given examples illustrate the nature of our assumptions.

Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function

Abstract In this article, we study the nonlinear and nonsmooth interval-valued optimization problems in the face of data uncertainty, which are called interval-valued robust optimization problems (IVROPs). We introduce the concept of nondominated solutions for the IVROP. If the interval-valued objective function f and constraint functions g i {g}_{i} are nonsmooth on Banach space E, we establish a nonsmooth and robust Karush-Kuhn-Tucker optimality theorem.

Robust Farkas-Minkowski Constraint Qualification for Convex Inequality System Under Data Uncertainty

The Farkas-Minkowski constraint qualification is an important concept within the theory and applications of mathematical programs with inequality constraints. In this paper, we mainly deal with the robust version of Farkas-Minkowski constraint qualification for convex inequality system under data uncertainty. We prove that the existence of robust global error bound is a sufficient condition for ensuring robust Farkas-Minkowski constraint qualification for convex inequality system in face of data uncertainty, where the uncertain data belong to a prescribed compact and convex uncertainty set. Moreover, we show that the converse is true for convex quadratic inequality system, when the uncertain data belong to a scenario uncertainty set.

Monetary Policy under Data Uncertainty: Interest-Rate Smoothing from a Cross-Country Perspective

Cross-country estimates of Taylor rules suggest that higher data uncertainty is associated with a more inertial behavior of interest rates. Data uncertainty is measured by the volatility of differences between real-time data and their revisions. Using a simple structural model with Kalman filter learning, I replicate the cross-country pattern of the inertial behavior. More inertial behavior results not because central banks gradually adjust interest rates in the face of data uncertainty, but because the central banks' inference about the true data is correlated with past interest rates. Thus, I endogenize the inertial behavior of interest rates as resulting in part from the learning process.

Sequential characterizations of robust optimal solutions in uncertain convex programs via perturbation approach

By using robust optimization approach, necessary and sufficient sequential optimality conditions without any constraint qualifications for the general convex optimization problem in the face of data uncertainty are given in terms of the ε-subdifferential. A sequential condition involving only the subdifferentials is also derived using a version of the Brøndsted-Rockafellar theorem. Consequently, sequential Lagrange multiplier condition for robust optimal solution of convex optimization problem with cone constraints in the face of data uncertainty is given. It is worth pointing out that there is no compactness assumption of uncertainty set and upper semicontinuity of functions involved in our results.

On approximate solutions for robust convex semidefinite optimization problems

In this paper, we consider approximate solutions ( $$\epsilon $$ -solutions) for a convex semidefinite programming problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we prove an approximate optimality theorem and approximate duality theorems for $$\epsilon $$ -solutions in robust convex semidefinite programming problem under the robust characteristic cone constraint qualification. Moreover, an example is given to illustrate the obtained results.

Approximate Optimality and Approximate Duality for Quasi Approximate Solutions in Robust Convex Semidefinite Programs

In this paper, we study quasi approximate solutions for a convex semidefinite programming problem in the face of data uncertainty. Using the robust optimization approach (worst-case approach), approximate optimality conditions and approximate duality theorems for quasi approximate solutions in robust convex semidefinite programming problems are explored under the robust characteristic cone constraint qualification. Moreover, some examples are given to illustrate the obtained results.

Some characterizations of robust optimal solutions for uncertain fractional optimization and applications

In this paper, following the framework of robust optimization, we consider robust optimal solutions for a fractional optimization problem in the face of data uncertainty both in the objective and constraints. To this end, by using the properties of the subdifferential sum formulae, we first introduce some robust basic subdifferential constraint qualifications, and then obtain some completely characterizations of the robust optimal solutions of this uncertain fractional optimization problem. We show that our results encompass as special cases some optimization problems considered in the recent literature. Moreover, as applications, the proposed approach is applied to investigate weakly robust efficient solutions for multi-objective fractional optimization problems in the face of data uncertainty both in the objective and constraints.

On quasi $$\epsilon $$ ϵ -solution for robust convex optimization problems

This paper devotes to the quasi \(\epsilon \)-solution (one sort of approximate solutions) for a robust convex optimization problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we establish approximate optimality theorem and approximate duality theorems in term of Wolfe type on quasi \(\epsilon \)-solution for the robust convex optimization problem. Moreover, some examples are given to illustrate the obtained results.

ON SOLUTION SET FOR CONVEX OPTIMIZATION PROBLEM WITH CONVEX INTEGRABLE OBJECTIVE FUNCTION AND GEOMETRIC CONSTRAINT SET

In this paper, we consider a convex optimization prob- lem with a convex integrable objective function and a geometric constraint set. We characterize the solution set of the problem when we know its one solution. 1. Introduction and preliminaries Convex optimization problems often have multiple solutions. Re- cently, Mangasarian (10) established simple and complete characteriza- tions for the solution set of the problem when we knew one solution of the problem. Since then, many authors have studied such charac- terizations for solution sets of several classes of optimization problems (2, 3, 5, 6, 8, 9, 11, 12, 13, 15). In particular, Jeyakumar, Lee and Dinh (5) showed that the Lagrangian function of a cone-constrained convex optimization problem, which has a cone-inequality constraint, is con- stant on its solution set, and then derived the Lagrange multiplier based characterizations of the solution set when we know one solution of the solution set. Moreover, Jeyakumar, Lee and Li (7) developed the char- acterizations of the solution sets to convex optimization problems in the face of data uncertainty.

Robust global error bounds for uncertain linear inequality systems with applications

In this paper, we examine global error bounds for a linear inequality system in the face of data uncertainty by building upon recent developments in robust optimization, where the uncertain parameters are assumed to be in prescribed uncertainty sets. We present necessary and sufficient dual conditions for the existence of robust global error bounds. As a consequence, we obtain a robust Hoffman error bound in the case of commonly used interval data uncertainty, extending the well-known Hoffman's error bound. We then introduce the notion of radius of robust global error bound under interval data uncertainty and present a formula for finding the radius by examining the stability of robust global error bounds. It provides the radius of the largest interval uncertainty set in which the uncertain system admits a robust global error bound. As an immediate application to robust linear programming, we provide conditions for robust feasibility and give a formula for radius of robust feasibility of an uncertain linear program.

Characterizing robust local error bounds for linear inequality systems under data uncertainty

We present necessary and sufficient conditions for the existence of robust local error bounds for linear inequality system in the face of data uncertainty where the uncertain data belong to a prescribed compact uncertainty set. The robust local error bound for an uncertain system is defined in terms of the existence of local error bound for its robust counterpart, where the uncertain linear inequality is enforced for every possible value of the data in the uncertainty set. Some of these conditions are expressed using the projection of the origin and others are given by way of normal cones at the boundary points of the solution set. In the case of commonly used interval data uncertainty, we show that a qualification condition completely characterizes the existence of robust local error bounds.

Some characterizations of robust optimal solutions for uncertain convex optimization problems

In this paper, we consider robust optimal solutions for a convex optimization problem in the face of data uncertainty both in the objective and constraints. By using the properties of the subdifferential sum formulae, we first introduce a robust-type subdifferential constraint qualification, and then obtain some completely characterizations of the robust optimal solution of this uncertain convex optimization problem. We also investigate Wolfe type robust duality between the uncertain convex optimization problem and its uncertain dual problem by proving duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. Moreover, we show that our results encompass as special cases some optimization problems considered in the recent literature.

OPTIMALITY CONDITIONS AND DUALITY IN NONDIFFERENTIABLE ROBUST OPTIMIZATION PROBLEMS

Abstract. We consider a nondi erentiable robust optimization problem,which has a maximum function of continuously di erentiable functionsand support functions as its objective function, continuously di erentiablefunctions as its constraint functions. We prove optimality conditions forthe nondi erentiable robust optimization problem. We formulate a Wolfetype dual problem for the nondi erentiable robust optimization problemand prove duality theorems. 1. IntroductionA standard form of nonlinear programming problem with inequality con-straints(P) inf x2R n ff(x) : g i (x) 0; i= 1; ;mg;where f: R n !R and g i : R n !R are continuously di erentiable functions.The problem in the face of data uncertainty in the constraints can be capturedby the following nonlinear programming problem:(UP) inf x2R n ff(x;u) : g i (x;v i ) 0; i= 1; ;mg;where u, v i are uncertain parameters and u2U, v i 2V i ;i= 1; ;mforsome convex compact sets U ˆR p ;V i ˆR q ;i= 1; ;m;respectively andf : R n nR p !R, g i : R R

On ϵ-solutions for robust fractional optimization problems

We consider ϵ-solutions (approximate solutions) for a fractional optimization problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we establish optimality theorems and duality theorems for ϵ-solutions for the fractional optimization problem. Moreover, we give an example illustrating our duality theorems. MSC:90C25, 90C32, 90C46.

Robust solutions to multi-objective linear programs with uncertain data

In this paper we examine multi-objective linear programming problems in the face of data uncertainty both in the objective function and the constraints. First, we derive a formula for radius of robust feasibility guaranteeing constraint feasibility for all possible uncertainties within a specified uncertainty set under affine data parametrization. We then present a complete characterization of robust weakly effcient solutions that are immunized against rank one objective matrix data uncertainty. We also provide classes of commonly used constraint data uncertainty sets under which a robust feasible solution of an uncertain multi-objective linear program can be numerically checked whether or not it is a robust weakly efficient solution.