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.     

Firefly penalty-based algorithm for bound constrained mixed-integer nonlinear programming

In this article, we aim to extend the firefly algorithm (FA) to solve bound constrained mixed-integer nonlinear programming (MINLP) problems. An exact penalty continuous formulation of the MINLP problem is used. The continuous penalty problem comes out by relaxing the integrality constraints and by adding a penalty term to the objective function that aims to penalize integrality constraint violation. Two penalty terms are proposed, one is based on the hyperbolic tangent function and the other on the inverse hyperbolic sine function. We prove that both penalties can be used to define the continuous penalty problem, in the sense that it is equivalent to the MINLP problem. The solutions of the penalty problem are obtained using a variant of the metaheuristic FA for global optimization. Numerical experiments are given on a set of benchmark problems aiming to analyze the quality of the obtained solutions and the convergence speed. We show that the firefly penalty-based algorithm compares favourably with the penalty algorithm when the deterministic DIRECT or the simulated annealing solvers are invoked, in terms of convergence speed.     

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.     

Analytic approximation and differentiability of joint chance constraints

ABSTRACT

An inner–outer approximation approach was recently developed to solve single chance constrained optimization (SCCOPT) problems. In this paper, we extend this approach to address joint chance constrained optimization (JCCOPT) problems. Using an inner–outer approximation, two smooth parametric optimization problems are defined whose feasible sets converge to the feasible set of JCCOPT from inside and outside, respectively. Any optimal solution of the inner approximation problem is a priori feasible to the JCCOPT. As the approximation parameter tends to zero, a subsequence of the solutions of the inner and outer problems, respectively, converge asymptotically to an optimal solution of the JCCOPT. As a main result, the continuous differentiability of the probability function of a joint chance constraint is obtained by examining the uniform convergence of the gradients of the parametric approximations.     

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.     

Algorithms for generalized potential games with mixed-integer variables

We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixed-integer variables, i.e., games in which some variables are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss–Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches.     

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.     

Convexity of chance constraints with independent random variables

We investigate the convexity of chance constraints with independent random variables. It will be shown, how concavity properties of the mapping related to the decision vector have to be combined with a suitable property of decrease for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels. It turns out that the required decrease can be verified for most prominent density functions. The results are applied then, to derive convexity of linear chance constraints with normally distributed stochastic coefficients when assuming independence of the rows of the coefficient matrix.     

Efficient and Adaptive Lagrange-Multiplier Methods for Nonlinear Continuous Global Optimization

Lagrangian methods are popular in solving continuous constrained optimization problems. In this paper, we address three important issues in applying Lagrangian methods to solve optimization problems with inequality constraints.First, we study methods to transform inequality constraints into equality constraints. An existing method, called the slack-variable method, adds a slack variable to each inequality constraint in order to transform it into an equality constraint. Its disadvantage is that when the search trajectory is inside a feasible region, some satisfied constraints may still pose some effect on the Lagrangian function, leading to possible oscillations and divergence when a local minimum lies on the boundary of the feasible region. To overcome this problem, we propose the MaxQ method that carries no effect on satisfied constraints. Hence, minimizing the Lagrangian function in a feasible region always leads to a local minimum of the objective function. We also study some strategies to speed up its convergence.Second, we study methods to improve the convergence speed of Lagrangian methods without affecting the solution quality. This is done by an adaptive-control strategy that dynamically adjusts the relative weights between the objective and the Lagrangian part, leading to better balance between the two and faster convergence.Third, we study a trace-based method to pull the search trajectory from one saddle point to another in a continuous fashion without restarts. This overcomes one of the problems in existing Lagrangian methods that converges only to one saddle point and requires random restarts to look for new saddle points, often missing good saddle points in the vicinity of saddle points already found.Finally, we describe a prototype Novel (Nonlinear Optimization via External Lead) that implements our proposed strategies and present improved solutions in solving a collection of benchmarks.     

New results for the Directed Profitable Rural Postman Problem

We study a generalization of the Directed Rural Postman Problem where not all arcs requiring a service have to be visited provided that a penalty cost is paid if a service arc is not crossed. The problem, known as Directed Profitable Rural Postman Problem, looks for a tour visiting the selected set of service arcs while minimizing both traveling and penalty costs. We add different valid inequalities to a known mathematical formulation of the problem and develop a branch-and-cut algorithm that introduces connectivity constraints both in a “lazy” and in a standard way. We also propose a matheuristic followed by an improvement heuristic (final refinement). The matheuristic exploits information provided by a problem relaxation to select promising service arcs used to solve optimally Directed Rural Postman problems. The ex-post refinement tries to improve the solution provided by the matheuristic using a branch-and-cut algorithm. The method gets a quick convergence through the introduction of connectivity cuts that are not guaranteed to be valid inequalities, and thus may exclude integer feasible solutions.     

Development of a new approach for deterministic supply chain network design

This paper proposes a mixed integer linear programming model and solution algorithm for solving supply chain network design problems in deterministic, multi-commodity, single-period contexts. The strategic level of supply chain planning and tactical level planning of supply chain are aggregated to propose an integrated model. The model integrates location and capacity choices for suppliers, plants and warehouses selection, product range assignment and production flows. The open-or-close decisions for the facilities are binary decision variables and the production and transportation flow decisions are continuous decision variables. Consequently, this problem is a binary mixed integer linear programming problem. In this paper, a modified version of Benders’ decomposition is proposed to solve the model. The most difficulty associated with the Benders’ decomposition is the solution of master problem, as in many real-life problems the model will be NP-hard and very time consuming. In the proposed procedure, the master problem will be developed using the surrogate constraints. We show that the main constraints of the master problem can be replaced by the strongest surrogate constraint. The generated problem with the strongest surrogate constraint is a valid relaxation of the main problem. Furthermore, a near-optimal initial solution is generated for a reduction in the number of iterations.     

A Path Algorithm for Constrained Estimation

Many least-square problems involve affine equality and inequality constraints. Although there are a variety of methods for solving such problems, most statisticians find constrained estimation challenging. The current article proposes a new path-following algorithm for quadratic programming that replaces hard constraints by what are called exact penalties. Similar penalties arise in l ₁ regularization in model selection. In the regularization setting, penalties encapsulate prior knowledge, and penalized parameter estimates represent a trade-off between the observed data and the prior knowledge. Classical penalty methods of optimization, such as the quadratic penalty method, solve a sequence of unconstrained problems that put greater and greater stress on meeting the constraints. In the limit as the penalty constant tends to ∞, one recovers the constrained solution. In the exact penalty method, squared penalties are replaced by absolute value penalties, and the solution is recovered for a finite value of the penalty constant. The exact path-following method starts at the unconstrained solution and follows the solution path as the penalty constant increases. In the process, the solution path hits, slides along, and exits from the various constraints. Path following in Lasso penalized regression, in contrast, starts with a large value of the penalty constant and works its way downward. In both settings, inspection of the entire solution path is revealing. Just as with the Lasso and generalized Lasso, it is possible to plot the effective degrees of freedom along the solution path. For a strictly convex quadratic program, the exact penalty algorithm can be framed entirely in terms of the sweep operator of regression analysis. A few well-chosen examples illustrate the mechanics and potential of path following. This article has supplementary materials available online.     

Exact penalty results for mathematical programs with vanishing constraints

A mathematical program with vanishing constraints (MPVC) is a constrained optimization problem arising in certain engineering applications. The feasible set has a complicated structure so that the most familiar constraint qualifications are usually violated. This, in turn, implies that standard penalty functions are typically non-exact for MPVCs. We therefore develop a new MPVC-tailored penalty function which is shown to be exact under reasonable assumptions. This new penalty function can then be used to derive (or recover) suitable optimality conditions for MPVCs.     

Optimal discrete-valued control computation

In this paper, we consider an optimal control problem in which the control takes values from a discrete set and the state and control are subject to continuous inequality constraints. By introducing auxiliary controls and applying a time-scaling transformation, we transform this optimal control problem into an equivalent problem subject to additional linear and quadratic constraints. The feasible region defined by these additional constraints is disconnected, and thus standard optimization methods struggle to handle these constraints. We introduce a novel exact penalty function to penalize constraint violations, and then append this penalty function to the objective. This leads to an approximate optimal control problem that can be solved using standard software packages such as MISER. Convergence results show that when the penalty parameter is sufficiently large, any local solution of the approximate problem is also a local solution of the original problem. We conclude the paper with some numerical results for two difficult train control problems.     

On generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables

ABSTRACT

We define and discuss different enumerative methods to compute solutions of generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables. We propose both branch-and-bound methods based on merit functions for the mixed-integer game, and branch-and-prune methods that exploit the concept of dominance to make effective cuts. We show that under mild assumptions the equilibrium set of the game is finite and we define an enumerative method to compute the whole of it. We show that our branch-and-prune method can be suitably modified in order to make a general equilibrium selection over the solution set of the mixed-integer game. We define an application in economics that can be modelled as a Nash game with linear coupling constraints and mixed-integer variables, and we adapt the branch-and-prune method to efficiently solve it.     

Concavity cuts for disjoint bilinear programming

We pursue the study of concavity cuts for the disjoint bilinear programming problem. This optimization problem has two equivalent symmetric linear maxmin reformulations, leading to two sets of concavity cuts. We first examine the depth of these cuts by considering the assumptions on the boundedness of the feasible regions of both maxmin and bilinear formulations. We next propose a branch and bound algorithm which make use of concavity cuts. We also present a procedure that eliminates degenerate solutions. Extensive computational experiences are reported. Sparse problems with up to 500 variables in each disjoint sets and 100 constraints, and dense problems with up to 60 variables again in each sets and 60 constraints are solved in reasonable computing times. Received: October 1999 / Accepted: January 2001?Published online March 22, 2001     

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.     

多变量、多约束连续或离散的非线性规划的一个通用算法

利用目标函数对约束函数关于设计变量的一阶微分或差分之比,给出了一个求解非线性规划的通用算法．不论变量和约束有多少,也不论变量是连续的还是离散的,这一算法都比较有效,尤其对离散非线性规划更有效．该方法是一种搜索法,勿需解任何数学方程,只需要计算函数值以及函数对变量的偏微分或差分值．许多数值例题和运筹学中一些经典问题,如1) 一、二维的背包问题;2) 一、二维资源分配问题;3) 复合系统工作可靠性问题;4) 机器负荷问题等,经用此法求解验证均较传统方法更有效和可靠．该方法的主要优点是:1) 不受问题的规模限制;2) 只要在可行域（集）内存在目标函数和约束函数及其一阶导数或差分的值,肯定可以搜索到最优的解,没有不收敛和不稳定的问题．     

一种具有非线性约束线性规划全局优化算法

本文提出了一种新的适用于处理非线性约束下线性规划问题的全局优化算法。该算法通过构造子问题来寻找优于当前局部最优解的可行解。该子问题可通过模拟退火算法来解决。通过求解一系列的子问题,当前最优解被不断地更新,最终求得全局最优解。最后,本算法应用于几个典型例题,并与罚函数法相比较,数值结果表明该算法是可行的,有效的。