a branch and cut decomposition algorithm for solving chance constrained mathematical programs with finite support we present a new approach for exactly solving chance constrained mathematical programs having discrete distributions with finite support and random polyhedral constraints such problems have been notoriously difficult to solve due to nonconvexity of the feasible region and most available methods are only able to find provably good solutions in certain very special cases our approach uses both decomposition to enable processing subproblems corresponding to one possible outcome at a time and integer programming techniques to combine the results of these subproblems to yield strong valid inequalities computational results on a chance constrained formulation of a resource planning problem inspired by a call center staffing application indicate the approach works significantly better than both an existing mixed integer programming formulation and a simple decomposition approach that does not use strong valid inequalities we also demonstrate how the approach can be used to efficiently solve for a sequence of risk levels as would be done when solving for the efficient frontier of risk and cost

tight formulations for some simple mixed integer programs and convex objective integer programs we study the polyhedral structure of simple mixed integer sets that generalize the two variable set s z r 1 z 1 s z b these sets form basic building blocks that can be used to derive tight formulations for more complicated mixed integer programs for four such sets we give a complete description by valid inequalities and or an integral extended formulation and we also indicate what constraints can be added without destroying integrality we apply these results to provide tight formulations for certain piecewise linear convex objective integer programs and in a companion paper we exploit them to provide polyhedral descriptions and computationally effective mixed integer programming formulations for discrete lot sizing problems

on intersection of two mixing sets with applications to joint chance constrained programs we study the polyhedral structure of a generalization of a mixing set described by the intersection of two mixing sets with two shared continuous variables where one continuous variable has a positive coefficient in one mixing set and a negative coefficient in the other our developments are motivated from a key substructure of linear joint chance constrained programs ccps with random right hand sides from a finite probability space the ccps of interest immediately admit a mixed integer programming reformulation nevertheless such standard reformulations are difficult to solve at large scale due to the weakness of their linear programming relaxations in this paper we initiate a systemic polyhedral study of such joint ccps by explicitly analyzing the system obtained from simultaneously considering two linear constraints inside the chance constraint we carry out our study on this particular intersection of two mixing sets under a nonnegativity assumption on data mixing inequalities are immediately applicable to our set yet they are not sufficient therefore we propose a new class of valid inequalities in addition to the mixing inequalities and establish conditions under which these inequalities are facet defining moreover under certain additional assumptions we prove that these new valid inequalities along with the classical mixing inequalities are sufficient in terms of providing the closed convex hull description of our set we also show that linear optimization over our set is polynomial time and we independently give a high order polynomial time separation algorithm for the new inequalities we complement our theoretical results with a computational study on the strength of the proposed inequalities our preliminary computational experiments with a fast heuristic separation approach demonstrate that our proposed inequalities are practically effective as well

bootstrapping a regression equation some empirical results abstract the bootstrap like the jackknife is a technique for estimating standard errors the idea is to use monte carlo simulation based on a nonparametric estimate of the underlying error distribution the main object of this article is to present the bootstrap in the context of an econometric equation describing the demand for energy by industry as it turns out the conventional asymptotic formulas for estimating standard errors are too optimistic by factors of nearly three when applied to a particular finite sample problem in a simpler context this finding can be given a mathematical proof

relaxations and approximations of chance constraints under finite distributions optimization problems with constraints involving stochastic parameters that are required to be satisfied with a prespecified probability threshold arise in numerous applications such chance constrained optimization problems involve the dual challenges of stochasticity and nonconvexity in the setting of a finite distribution of the stochastic parameters an optimization problem with linear chance constraints can be formulated as a mixed integer linear program milp the natural milp formulation has a weak relaxation bound and is quite difficult to solve in this paper we review some recent results on improving the relaxation bounds and constructing approximate solutions for milp formulations of chance constraints we also discuss a recently introduced bicriteria approximation algorithm for covering type chance constrained problems this algorithm uses a relaxation to construct a solution whose constraint violation risk level may be larger than the pre specified threshold but is within a constant factor of it and whose objective value is also within a constant factor of the true optimal value finally we present some new results that improve on the bicriteria approximation factors in the finite scenario setting and shed light on the effect of strong relaxations on the approximation ratios

on quantile cuts and their closure for chance constrained optimization problems a chance constrained optimization problem over a finite distribution involves a set of scenario constraints from which a small subset can be violated we consider the setting where all scenario constraints are mixed integer convex existing works typically consider a mixed integer nonlinear programming minlp formulation of this problem by introducing binary variables to indicate which constraint systems are to be satisfied or violated a variety of cutting plane approaches for this minlp formulation have been developed in this paper we consider a family of cuts in the original space rather than those in the extended space of the minlp reformulation these cuts known as quantile cuts can be viewed as a projection of the well known family of mixing inequalities for the minlp reformulation onto the original problem space we show that the closure of the infinite family of all quantile cuts has a finite description an important corollary of this result is that for linear chance constrained problems the quantile closure is polyhedral we further show that a recursive application of quantile closure operations recovers the convex hull of the nonconvex chance constrained set in the limit and in the pure integer setting the convergence is finite we show that separation of quantile cuts is in general np hard develop a heuristic separation method and demonstrate its effectiveness through a computational study we also study an approximation of the quantile closure and propose a generalization by grouping scenarios

boundary behavior of interior point algorithms in linear programming this paper studies the boundary behavior of some interior point algorithms for linear programming the algorithms considered are karmarkar s projective rescaling algorithm the linear rescaling algorithm which was proposed as a variation on karmarkar s algorithm and the logarithmic barrier technique the study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits it is shown that although the algorithms are defined on the interior of the feasible polyhedron they actually determine differentiable vector fields on the closed polyhedron conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the klee minty cube the linear rescaling algorithm satisfies these conditions thus limits of such trajectories obtained when a starting point is pushed to the boundary may have an exponential number of breakpoints it is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints however projective rescaling trajectories may visit the neighborhoods of linearly many vertices the behavior of the linear rescaling algorithm near vertices is analyzed it is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum

efficiency and robustness in resampling it is known that the standard delete 1 jackknife and the classical bootstrap are in general equally efficient for estimating the mean square error of a statistic in the i i d setting however this equivalence no longer holds in the linear regression model it turns out that the bootstrap is more efficient when error variables are homogeneous and the jackknife is more robust when they are heterogeneous in fact we can divide all the commonly used resampling procedures for linear regression models into two types the e type the efficient ones like the bootstrap and the r type the robust ones like the jackknife thus the theory presented here provides a unified view of all the known resampling procedures in linear regression

decomposition algorithms for two stage chance constrained programs we study a class of chance constrained two stage stochastic optimization problems where second stage feasible recourse decisions incur additional cost in addition we propose a new model where recovery decisions are made for the infeasible scenarios to obtain feasible solutions to a relaxed second stage problem we develop decomposition algorithms with specialized optimality and feasibility cuts to solve this class of problems computational results on a chance constrained resource planing problem indicate that our algorithms are highly effective in solving these problems compared to a mixed integer programming reformulation and a naive decomposition method

generalized bootstrap for estimating equations we introduce a generalized bootstrap technique for estimators obtained by solving estimating equations some special cases of this generalized bootstrap are the classical bootstrap of efron the delete d jackknife and variations of the bayesian bootstrap the use of the proposed technique is discussed in some examples distributional consistency of the method is established and an asymptotic representation of the resampling variance estimator is obtained

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

mixing mixed integer inequalities mixed integer rounding mir inequalities play a central role in the development of strong cutting planes for mixed integer programs in this paper we investigate how known mir inequalities can be combined in order to generate new strong valid inequalities given a mixed integer region s and a collection of valid base mixed integer inequalities we develop a procedure for generating new valid inequalities for s the starting point of our procedure is to consider the mir inequalities related with the base inequalities for any subset of these mir inequalities we generate two new inequalities by combining or mixing them we show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed integer region associated with the base inequalities we discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed integer programming problems in particular we present examples for production planning capacitated facility location capacitated network design and multiple knapsack problems we also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs finally we study some extensions of this mixing procedure

concavity and efficient points of discrete distributions in probabilistic programming we consider stochastic programming problems with probabilistic constraints involving integer valued random variables the concept of a p efficient point of a probability distribution is used to derive various equivalent problem formulations next we introduce the concept of r concave discrete probability distributions and analyse its relevance for problems under consideration these notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables the results are illustrated with numerical examples

on safe tractable approximations of chance constraints abstract a natural way to handle optimization problem with data affected by stochastic uncertainty is to pass to a chance constrained version of the problem where candidate solutions should satisfy the randomly perturbed constraints with probability at least 1 while being attractive from modeling viewpoint chance constrained problems as they are are in general computationally intractable in this survey paper we overview several simulation based and simulation free computationally tractable approximations of chance constrained convex programs primarily those of chance constrained linear conic quadratic and semidefinite programming

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

the mixed vertex packing problem we study a generalization of the vertex packing problem having both binary and bounded continuous variables called the mixed vertex packing problem mvpp the well known vertex packing model arises as a subproblem or relaxation of many 0 1 integer problems whereas the mixed vertex packing model arises as a natural counterpart of vertex packing in the context of mixed 0 1 integer programming we describe strong valid inequalities for the convex hull of solutions to the mvpp and separation algorithms for these inequalities we give a summary of computational results with a branch and cut algorithm for solving the mvpp and using it to solve general mixed integer problems

a polyhedral study on chance constrained program with random right hand side the essential structure of the mixed integer programming formulation for chance constrained program ccp is the intersection of multiple mixing sets with a 0 1 knapsack to improve our computational capacity on ccp an underlying substructure the single mixing set with a 0 1 knapsack has received substantial attentions recently in this study we consider a ccp problem with stochastic right hand side under a finite discrete distribution we first present a family of strong inequalities that subsumes known facet defining ones for that single mixing set due to the flexibility of our generalized inequalities we develop a new separation heuristic that has a complexity much less than existing one and guarantees generated cutting planes are facet defining for the polyhedron of ccp then we study lifting and superadditive lifting on knapsack cover inequalities and provide an implementable procedure on deriving another family of strong inequalities for the single mixing set finally different from the traditional approach that aggregates original constraints to investigate polyhedral implications due to their interactions we propose a novel blending procedure that produces strong valid inequalities for ccp by integrating those derived from individual mixing sets we show that under certain conditions they are the first type of facet defining inequalities describing intersection of multiple mixing sets and design an efficient separation heuristic for implementation in the computational experiments we perform a systematic study and illustrate the efficiency of the proposed inequalities on solving chance constrained static probabilistic lot sizing problems

an integer programming approach for linear programs with probabilistic constraints linear programs with joint probabilistic constraints pclp are difficult to solve because the feasible region is not convex we consider a special case of pclp in which only the right hand side is random and this random vector has a finite distribution we give a mixed integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint we obtain two strengthened formulations as a byproduct of this analysis we obtain new results for the previously studied mixing set subject to an additional knapsack inequality we present computational results which indicate that by using our strengthened formulations instances that are considerably larger than have been considered before can be solved to optimality

on the mixing set with a knapsack constraint we study a substructure appearing in mixed integer programming reformulations of chance constrained programs with stochastic right hand sides over a finite discrete distribution which we call the mixing set with a knapsack constraint recently luedtke et al math program 122 2 247 272 2010 and kucukyavuz math program 132 1 31 56 2012 studied valid inequalities for such sets however most of their results were focused on the equal probabilities case when the knapsack constraint reduces to a cardinality constraint in this paper we focus on the general probabilities case general knapsack constraint we characterize the valid inequalities that do not come from the knapsack polytope and use this characterization to generalize the results previously derived for the equal probabilities case our results allow for a deep understanding of the relationship that the set under consideration has with the knapsack polytope moreover they allow us to establish benchmarks that can be used to identify when a relaxation will be useful for the considered types of reformulations of chance constrained programs

convex approximations of chance constrained programs we consider a chance constrained problem where one seeks to minimize a convex objective over solutions satisfying with a given close to one probability a system of randomly perturbed convex constraints this problem may happen to be computationally intractable our goal is to build its computationally tractable approximation i e an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem we construct a general class of such convex conservative approximations of the corresponding chance constrained problem moreover under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables we build a large deviation type approximation referred to as bernstein approximation of the chance constrained problem this approximation is convex and efficiently solvable we propose a simulation based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the bernstein and well known scenario approximation approaches finally we extend our construction to the case of ambiguous chance constrained problems where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly while the chance constraint should be satisfied for every distribution given by this set

a sample approximation approach for optimization with probabilistic constraints we study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample we show that such a sample approximation problem with a risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast this leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound we then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability once again we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem finally we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints

on mixing sets arising in chance constrained programming the mixing set with a knapsack constraint arises in deterministic equivalent of chance constrained programming problems with finite discrete distributions we first consider the case that the chance constrained program has equal probabilities for each scenario we study the resulting mixing set with a cardinality constraint and propose facet defining inequalities that subsume known explicit inequalities for this set we extend these inequalities to obtain valid inequalities for the mixing set with a knapsack constraint in addition we propose a compact extended reformulation with polynomial number of variables and constraints that characterizes a linear programming equivalent of a single chance constraint with equal scenario probabilities we introduce a blending procedure to find valid inequalities for intersection of multiple mixing sets we propose a polynomial size extended formulation for the intersection of multiple mixing sets with a knapsack constraint that is stronger than the original mixing formulation we also give a compact extended linear program for the intersection of multiple mixing sets and a cardinality constraint for a special case we illustrate the effectiveness of the proposed inequalities in our computational experiments with probabilistic lot sizing problems

solving chance constrained optimization problems with stochastic quadratic inequalities we propose a new and systematic reformulation and algorithmic approach to solve a complex class of stochastic programming problems involving a joint chance constraint with random technology matrix and stochastic quadratic inequalities the method is general enough to apply to nonconvex as well as nonseparable quadratic terms we derive two new reformulations and give sufficient conditions under which the reformulated problem is equivalent the second reformulation provides a much sparser representation of the feasible set of the chance constraint and offers tremendous computational advantages this new reformulation method can be used for linear stochastic inequalities and will also significantly improve the solution of such joint chance constrained problems we provide general and easily identifiable conditions under which the base reformulations can be linearized we show that the size of the reformulated problems in particular their number of binary variables and quadratic mixed integer terms does n

nonanticipative duality relaxations and formulations for chance constrained stochastic programs we propose two new lagrangian dual problems for chance constrained stochastic programs based on relaxing nonanticipativity constraints we compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed integer programming mip formulation for a given dual solution the associated lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem we also derive two new primal mip formulations and demonstrate that for chance constrained linear programs the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds we propose a new heuristic method and two new exact algorithms based on these duals and formulations the first exact algorithm applies to chance constrained binary programs and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems the second exact method is a branch and cut algorithm for solving either of the primal formulations our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently and the gaps between the best dual bounds and the heuristic solutions are small