Convexity of chance constrained programming problems with respect to a new generalized concavity notion

In this paper, convexity of chance constrained problems have been investigated. A new generalization of convexity concept, named h-concavity, has been introduced and it has been shown that this new concept is the generalization of the ??-concavity. Then, using the new concept, some of the previous results obtained by Shapiro et al. [in Lecture Notes on Stochastic Programming Modeling and Theory, SIAM and MPS, 2009] on properties of ??-concave functions, have been extended. Next the convexity of chance constraints with independent random variables is investigated. It will be shown how concavity properties of the mapping related to the decision vector have to be combined with suitable properties of decrease or increase for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels and then sufficient conditions for convexity of chance constrained problems which has been introduced by Henrion and Strugarek [in Convexity of chance constraints with independent random variables. Comput. Optim. Appl. 41:263?C276, 2008] has been extended in this paper for a wider class of real functions.     

Eventual convexity of chance constrained feasible sets

In decision-making problems where uncertainty plays a key role and decisions have to be taken prior to observing uncertainty, chance constraints are a strong modelling tool for defining safety of decisions. These constraints request that a random inequality system depending on a decision vector has to be satisfied with a high probability. The characteristics of the feasible set of such chance constraints depend on the constraint mapping of the random inequality system, the underlying law of uncertainty and the probability level. One characteristic of particular interest is convexity. Convexity can be shown under fairly general conditions on the underlying law of uncertainty and on the constraint mapping, regardless of the probability-level. In some situations, convexity can only be shown when the probability-level is high enough. This is defined as eventual convexity. In this paper, we will investigate further how eventual convexity can be assured for specially structured chance constraints involving Copulae. The Copulae have to exhibit generalized concavity properties. In particular, we will extend recent results and exhibit a clear link between the generalized concavity properties of the various mappings involved in the chance constraint for the result to hold. Various examples show the strength of the provided generalization.     

Characterization of Stability for Cone Increasing Constraint Mappings

We investigate stability (in terms of metric regularity) for the specific class of cone increasing constraint mappings. This class is of interest in problems with additional knowledge on some nondecreasing behavior of the constraints (e.g. in chance constraints, where the occurring distribution function of some probability measure is automatically nondecreasing). It is demonstrated, how this extra information may lead to sharper characterizations. In the first part, general cone increasing constraint mappings are studied by exploiting criteria for metric regularity, as recently developed by Mordukhovich. The second part focusses on genericity investigations for global metric regularity (i.e. metric regularity at all feasible points) of nondecreasing constraints in finite dimensions. Applications to chance constraints are given.     

Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization

Chance constraints represent a popular tool for finding decisions that enforce the satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in finite dimensions. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties as well as a stability result to an infinite dimensional setting. The abstract results are applied to a simple PDE constrained control problem subject to (uniform) state chance constraints.     

Stochastic programming problems with generalized integrated chance constraints

If the constraints in an optimization problem are dependent on a random parameter, we would like to ensure that they are fulfilled with a high level of reliability. The most natural way is to employ chance constraints. However, the resulting problem is very hard to solve. We propose an alternative formulation of stochastic programs using penalty functions. The expectations of penalties can be left as constraints leading to generalized integrated chance constraints, or incorporated into the objective as a penalty term. We show that the penalty problems are asymptotically equivalent under quite mild conditions. We discuss applications of sample-approximation techniques to the problems with generalized integrated chance constraints and propose rates of convergence for the set of feasible solutions. We will direct our attention to the case when the set of feasible solutions is finite, which can appear in integer programming. The results are then extended to the bounded sets with continuous variables. Additional binary variables are necessary to solve sample-approximated chance-constrained problems, leading to a large mixed-integer non-linear program. On the other hand, the problems with penalties can be solved without adding binary variables; just continuous variables are necessary to model the penalties. The introduced approaches are applied to the blending problem leading to comparably reliable solutions.     

Convexity and optimization with copulæ structured probabilistic constraints

Probability constraints play a key role in optimization problems involving uncertainties. These constraints request that an inequality system depending on a random vector has to be satisfied with a high enough probability. In specific settings, copulæ can be used to model the probabilistic constraints with uncertainty on the left-hand side. In this paper, we provide eventual convexity results for the feasible set of decisions under local generalized concavity properties of the constraint mappings and involved copulæ. The results cover all Archimedean copulæ. We consider probabilistic constraints wherein the decision and random vector are separated, i.e. left/right-hand side uncertainty. In order to solve the underlying optimization problem, we propose and analyse convergence of a regularized supporting hyperplane method: a stabilized variant of generalized Benders decomposition. The algorithm is tested on a large set of instances involving several copulæ among which the Gaussian copula. A Numerical comparison with a (pure) supporting hyperplane algorithm and a general purpose solver for non-linear optimization is also presented.     

On Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Equilibrium Constraints

In this article, necessary conditions of Fritz John type for weak efficient solutions of a nonsmooth vector equilibrium problem involving equilibrium constraints (VEPEC) in terms of the Clarke subdifferentials are established. Under constraint qualifications which are suitable for (VEPEC), necessary conditions of Kuhn-Tucker type for efficiency are derived. Under assumptions on generalized convexity of data, sufficient conditions for efficiency are developed. Some applications to vector variational inequalities and vector optimization problems with equilibrium constraints are also given.     

Convergent Bounds for Stochastic Programs with Expected Value Constraints

This article describes a bounding approximation scheme for convex multistage stochastic programs (MSP) that constrain the conditional expectation of some decision-dependent random variables. Expected value constraints of this type are useful for modelling a decision maker’s risk preferences, but they may also arise as artifacts of stage-aggregation. We develop two finite-dimensional approximate problems that provide bounds on the (infinite-dimensional) original problem, and we show that the gap between the bounds can be made smaller than any prescribed tolerance. Moreover, the solutions of the approximate MSPs give rise to a feasible policy for the original MSP, and this policy’s optimality gap is shown to be smaller than the difference of the bounds. The considered problem class comprises models with integrated chance constraints and conditional value-at-risk constraints. No relatively complete recourse is assumed.     

Reliability stochastic optimization for a series system with interval component reliability via genetic algorithm

This paper deals with chance constraints based reliability stochastic optimization problem in the series system. This problem can be formulated as a nonlinear integer programming problem of maximizing the overall system reliability under chance constraints due to resources. The assumption of traditional reliability optimization problem is that the reliability of a component is known as a fixed quantity which lies in the open interval (0, 1). However, in real life situations, the reliability of an individual component may vary due to some realistic factors and it is sensible to treat this as a positive imprecise number and this imprecise number is represented by an interval valued number. In this work, we have formulated the reliability optimization problem as a chance constraints based reliability stochastic optimization problem with interval valued reliabilities of components. Then, the chance constraints of the problem are converted into the equivalent deterministic form. The transformed problem has been formulated as an unconstrained integer programming problem with interval coefficients by Big-M penalty technique. Then to solve this problem, we have developed a real coded genetic algorithm (GA) for integer variables with tournament selection, uniform crossover and one-neighborhood mutation. To illustrate the model two numerical examples have been solved by our developed GA. Finally to study the stability of our developed GA with respect to the different GA parameters, sensitivity analyses have been done graphically.     

A linear approach to shape preserving spline approximation

This paper deals with the approximation of a given large scattered univariate or bivariate data set that possesses certain shape properties, such as convexity, monotonicity, and/or range restrictions. The data are approximated for instance by tensor-product B-splines preserving the shape characteristics present in the data.Shape preservation of the spline approximant is obtained by additional linear constraints. Constraints are constructed which are local linear sufficient conditions in the unknowns for convexity or monotonicity. In addition, it is attractive if the objective function of the resulting minimisation problem is also linear, as the problem can then be written as a linear programming problem. A special linear approach based on constrained least squares is presented, which in the case of large data reduces the complexity of the problem sets in contrast with that obtained for the usual ₂-norm as well as the -norm.An algorithm based on iterative knot insertion which generates a sequence of shape preserving approximants is given. It is investigated which linear objective functions are suited to obtain an efficient knot insertion method.     

Energy and reserve dispatch with distributionally robust joint chance constraints

We develop a two-stage stochastic program for energy and reserve dispatch of a joint power and gas system with a high penetration of renewables. Data-driven distributionally robust chance constraints ensure that there is no load shedding and renewable spillage with high probability. We solve this problem efficiently using conditional value-at-risk approximations and linear decision rules. Out-of-sample experiments show that this model dominates the corresponding stochastic program without chance constraints that models the effects of load shedding and renewable spillage explicitly.     

Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations. We give a simple proof to the M-stationary condition and show that it is sufficient for global or local optimality under some MPEC generalized convexity assumptions. Moreover, we propose new constraint qualifications for M-stationary conditions to hold. These new constraint qualifications include piecewise MFCQ, piecewise Slater condition, MPEC weak reverse convex constraint qualification, MPEC Arrow-Hurwicz-Uzawa constraint qualification, MPEC Zangwill constraint qualification, MPEC Kuhn-Tucker constraint qualification, and MPEC Abadie constraint qualification.     

Some kind of Pareto stationarity for multiobjective problems with equilibrium constraints

ABSTRACT

In this paper, we derive some optimality and stationarity conditions for a multiobjective problem with equilibrium constraints (MOPEC). In particular, under a generalized Guignard constraint qualification, we show that any locally Pareto optimal solution of MOPEC must satisfy the strong Pareto Kuhn-Tucker optimality conditions. We also prove that the generalized Guignard constraint qualification is the weakest constraint qualification for the strong Pareto Kuhn-Tucker optimality. Furthermore, under certain convexity or generalized convexity assumptions, we show that the strong Pareto Kuhn-Tucker optimality conditions are also sufficient for several popular locally Pareto-type optimality conditions for MOPEC.     

A model for dynamic chance constraints in hydro power reservoir management

In this paper, a model for (joint) dynamic chance constraints is proposed and applied to an optimization problem in water reservoir management. The model relies on discretization of the decision variables but keeps the probability distribution continuous. Our approach relies on calculating probabilities of rectangles which is particularly useful in the presence of independent random variables but works equally well in the case of correlated variables. Numerical results are provided for two and three stages.     

Dirichlet Boundary Control of Hyperbolic Equations in the Presence of State Constraints

We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems.     

具有模糊三角要素的机会约束型DEA模型

确定性 DEA是一种数据敏感的评价方法 ,在某些不利情况下 ,数据的轻微变动将极大的影响评价结果 .针对这一不足 ,根据模糊机会约束规划的理论框架 ,基于 C2 R模型 ,建立了具有模糊三角要素的机会约束型 DEA模型 ( FCCPDEA) ,用机会约束来描述这一不确定性 .在 FCCPDEA模型中 ,从令决策者满意的角度出发 ,用决策者期望各决策单元能达到的最大效率 ,即效率基数 ,来代替理想效率 1,使评价更符合现实 .     

一类带概率互补约束的随机优化问题的最优性条件

主要讨论了一类带概率互补约束的随机优化问题的最优性条件.首先利用一类非线性互补(NCP)函数将概率互补约束转化成为一个通常的概率约束.然后,利用概率约束的相关理论结果,将其等价地转化成一个带不等式约束的优化问题.最后给出了这类问题的弱驻点和最优解的最优性条件.     

On stability in multiobjective programming — A stochastic approach

In this paper we assume that a deterministic multiobjective programming problem is approximated by surrogate problems based on estimations for the objective functions and the constraints. Making use of a large deviations approach, we investigate the behaviour of the constraint sets, the sets of efficient points and the solution sets if the size of the underlying sample tends to infinity. The results are illustrated by applying them to stochastic programming with chance constraints, where (i) the distribution function of the random variable is estimated by the empirical distribution function, (ii) certain parameters have to be estimated.     

QPECgen, a MATLAB Generator for Mathematical Programs with Quadratic Objectives and Affine Variational Inequality Constraints

We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, first-level constraints are linear, and second-level (equilibrium) constraints are given by a parametric affine variational inequality or one of its specialisations. The generator, written in MATLAB, allows the user to control different properties of the QPEC and its solution. Options include the proportion of degenerate constraints in both the first and second level, ill-conditioning, convexity of the objective, monotonicity and symmetry of the second-level problem, and so on. We believe these properties may substantially effect efficiency of existing methods for MPEC, and illustrate this numerically by applying several methods to generator test problems. Documentation and relevant codes can be found by visiting http://www.ms.unimelb.edu.au/danny/qpecgendoc.html.