We consider a class of stochastic mathematical programs with complementarity constraints, in which both the objective and the constraints involve limit functions or expectations that need to be estimated or approximated. Such programs can be used for modeling \\average" or steady-state behavior of complex stochastic systems. Recently, simulation-based methods have been successfully used for solving challenging stochastic optimization problems and equilibrium models. Here we broaden the applicability of so-called the sample-path method to include the solution of certain stochastic mathematical programs with equilibrium constraints. The convergence analysis of sample-path methods rely heavily on stability conditions. We first review necessary sensitivity results, then describe the method, and provide sufficient conditions for its almost-sure convergence. Alongside we provide a complementary sensitivity result for the corresponding deterministic problems. In addition, we also provide a unifying discussion on alternative set of sufficient conditions, derive a complementary result regarding the analysis of stochastic variational inequalities, and prove the equivalence of two different regularity conditions.simulation;mathematical programs with equilibrium constraints;stability;regularity conditions;sample-path methods;stochastic mathematical programs with complementarity constraints

This chapter presents a self-contained approach of variational analysis and generalized differentiation to deriving necessary optimality in problems of bilevel optimization with Lipschitzian data. We mainly concentrate on optimistic models, although the developed machinery also applies to pessimistic versions. Some open problems are posed and discussed.

In this paper we introduce notions of well-posedness for a vector optimization problem and for a vector variational inequality of differential type, we study their basic properties and we establish the links among them. The proposed concept of well-posedness for a vector optimization problem generalizes the notion of well-setness for scalar optimization problems, introduced in [2]. On the other side, the introduced definition of well-posedness for a vector variational inequality extends the one given in [13] for the scalar case.Keywords: vector optimization, vector variational inequality, well-posedness

Due to lack of data, overfitting ubiquitously exists in real-world applications of deep neural networks (DNNs). We propose advanced dropout, a model-free methodology, to mitigate overfitting and improve the performance of DNNs. The advanced dropout technique applies a model-free and easily implemented distribution with parametric prior, and adaptively adjusts dropout rate. Specifically, the distribution parameters are optimized by stochastic gradient variational Bayes in order to carry out an end-to-end training. We evaluate the effectiveness of the advanced dropout against nine dropout techniques on seven computer vision datasets (five small-scale datasets and two large-scale datasets) with various base models. The advanced dropout outperforms all the referred techniques on all the datasets.We further compare the effectiveness ratios and find that advanced dropout achieves the highest one on most cases. Next, we conduct a set of analysis of dropout rate characteristics, including convergence of the adaptive dropout rate, the learned distributions of dropout masks, and a comparison with dropout rate generation without an explicit distribution. In addition, the ability of overfitting prevention is evaluated and confirmed. Finally, we extend the application of the advanced dropout to uncertainty inference, network pruning, text classification, and regression. The proposed advanced dropout is also superior to the corresponding referred methods. Codes are available at https://github.com/PRIS-CV/AdvancedDropout.

Let E be a linear space, K E and f : K ? R. We put in terms of the lower Dini directional derivative a problem, referred to as GMV I(f ,K), which can be considered as a generalization of the Minty variational inequality of differential type (for short, MV I(f ,K)). We investigate, in the case of K star-shaped (for short, st-sh), the existence of a solution x of GMV I(f ,K) and the property of f to increase-along-rays starting at x (for short, f IAR(K, x )). We prove that GMV I(f ,K) with radially l.s.c. function f has a solution x ker K if and only if f IAR(K, x ). Further, we prove, that the solution set of GMV I(f ,K) is a convex and radially closed subset of kerK. We show also that, if GMV I(f ,K) has a solution x K, then x is a global minimizer of the problem f(x) ? min, x K. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove, that in case of a quasi-convex function f, these sets coincide. Key words: Minty variational inequality, Generalized variational inequality, Existence of solutions, Increase along rays, Quasi-convex functions.

In this paper we extend to the vector case the notion of increasing along rays function. The proposed definition is given by means of a nonlinear scalarization through the so-called oriented distance function from a point to a set. We prove that the considered class of functions enjoys properties similar to those holding in the scalar case, with regard to optimization problems, relations with (generalized) convex functions and characterization in terms of Minty type variational inequalities. Key words: generalized convexity, increase-along-rays property, star-shaped set, Minty variational inequality.

In this paper, we suggest and consider a class of new three-step approximation schemes for generalized general set-valued mixed quasi variational inequalities. We also consider and analyze a new class of extragradient-type methods for solving generalized set-valued variational inequalities. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems

[[abstract]]In this paper, we study the well-posedness in the generalized sense for variational inclusion problems and variational disclusion problems, the well-posedness for optimization problems with variational inclusion problems, variational disclusion problems and scalar equilibrium problems as constraint

In this paper we consider a vector optimization problem; we present some scalarization techniques for finding all the vector optimal points of this problem and we discuss the relationships between these methods. Moreover, in the linear case, the study of dual variables is carried on by means of sensitivity analysis and also by a parametric approach. We also give an interpretation of the dual variables as marginal rates of substitution of an objective function with respect to another one, and of an objective function with respect to a constraint.Vector Optimization, Image Space, Separation, Scalarization, Shadow Prices

The maximum (or minimum) generalized eigenvalue of symmetric positive semidefinite matrices that depend on optimization variables often appears as objective or constraint functions in structural topology optimization when we consider robustness, vibration, and buckling. It can be an unbounded or discontinuous function where matrices become singular (where a topological change of the structural design occurs). Based on variational analysis, we redefine the maximum (and minimum) generalized eigenvalue function as an extended real-valued function and propose a real-valued continuous approximation of it. Then, we show that the proposed approximation epi-converges to the original redefined function, which justifies solving problems with the approximation instead. We consider two specific topology optimization problems: robust compliance optimization and eigenfrequency optimization. Although we conduct numerical experiments only on truss topology optimization, the main theoretical results can also be applied to topology optimization of continua.

inequality;simulation;optimization

A surface parameterization is a function that maps coordinates in a 2-dimensional parameter space to points on a surface. This thesis investigates two kinds of parameterizations for surfaces that are disc-like in shape. The first is a map from a region of the plane to the surface. The second is a mapping from one surface to another, which defines a correspondence between them. The main challenge in both cases is the construction of a smooth map with low distortion. In this thesis we present a variational approach to surface parameterization that addresses these challenges. The first contribution of this thesis is the development of a variational framework for parameterizations. This framework encompasses the mapping of a region of the plane to a surface that is isomorphic to a disc, and the mapping between such surfaces. It is based on the rich mathematical theory built up over decades in the study of rational mechanics. Because of its roots in mechanics, our parameterizations are guaranteed to be smooth and locally bijective, and optimal parameterizations which minimize a variational energy are known to exist. A proof of existence is given for the case of optimal parameterizations in the plane. Our second contribution is a set of algorithms to construct parameterizations for surface triangulations. We describe in detail free-boundary methods that use standard numerical optimization algorithms for the computation of optimal parameterizations. A flexible set of parameters is offered to the user to formulate preferences for the trade-off between angle, area and length distortion in parameterizations in the plane. In the specification of a correspondence between surfaces, we provide user control through feature lines which are mapped as sets onto corresponding feature lines. Additionally we allow for a partial correspondence of the surfaces which is particularly important for correlating surfaces with boundaries. Our third contribution is an analysis of the performance of the algorithms based on our implementations. Our testing focuses on parameterizations of physically-acquired triangle mesh data. The efficiency of our methods is measured by analyzing the rate of convergence of the energy minimization, and execution times are shown to be quite reasonable. Robustness is established in the presence of large deformations in the parameterizations and the stability of our parameterizations is demonstrated under different discretizations of the surfaces. Our fourth contribution is a set of concrete, compelling applications of surface parameterization. Non-trivial examples which draw from texture mapping, morphing and facial animation provide further evidence and insight into the versatility of our parameterization framework.</p