validation analysis of mirror descent stochastic approximation method the main goal of this paper is to develop accuracy estimates for stochastic programming problems by employing stochastic approximation sa type algorithms to this end we show that while running a mirror descent stochastic approximation procedure one can compute with a small additional effort lower and upper statistical bounds for the optimal objective value we demonstrate that for a certain class of convex stochastic programs these bounds are comparable in quality with similar bounds computed by the sample average approximation method while their computational cost is considerably smaller

convex approximations for two stage mixed integer mean risk recourse models with conditional value at risk in traditional two stage mixed integer recourse models the expected value of the total costs is minimized in order to address risk averse attitudes of decision makers we consider a weighted mean risk objective instead conditional value at risk is used as our risk measure integrality conditions on decision variables make the model non convex and hence hard to solve to tackle this problem we derive convex approximation models and corresponding error bounds that depend on the total variations of the density functions of the random right hand side variables in the model we show that the error bounds converge to zero if these total variations go to zero in addition for the special cases of totally unimodular and simple integer recourse models we derive sharper error bounds

on a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling we consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation we study a sample average approximation saa approach that uses nested sampling in which at each stage a number of scenarios are examined and a number of simulation replications are performed for each scenario to estimate the next stage constraints this approach provides an approximate solution to the multistage problem to establish the consistency of the saa approach we first consider a two stage problem and show that in the second stage problem given a scenario the optimal values and solutions of the saa converge to those of the true problem with probability one when the sample sizes go to infinity these convergence results do not hold uniformly over all possible scenarios for the second stage problem we are nevertheless able to prove that the optimal values and solutions of the saa converge to the true ones with probability one when the sample sizes at both stages increase to infinity we also prove exponential convergence of the probability of a large deviation for the optimal value of the saa the true value of an optimal solution of the saa and the probability that any optimal solution to the saa is an optimal solution of the true problem all of these results can be extended to a multistage setting and we explain how to do it our framework and saa results cover a large variety of resource allocation problems for which at each stage after the first one new information becomes available and the allocation can be readjusted under constraints that involve expectations estimated by monte carlo as an illustration we apply this saa method to a staffing problem in a call center in which the goal is to optimize the numbers of agents of each type under some constraints on the quality of service qos the staffing allocation has to be decided under an uncertain arrival rate with a prior distribution in the first stage and can be adjusted at some additional cost when better information on the arrival rate becomes available in later stages

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

semipositone problems with falling zeros on exterior domains abstract we study boundary value problems of the form u k x f u x u 0 if x r 0 u 0 as x where is a positive parameter u div u is the laplacian of u x r n n 2 x r 0 k belongs to a class of c 1 functions such that lim r k r 0 and f belongs to a class of c 1 functions which are negative at the origin and have falling zeros we discuss the existence and uniqueness of nonnegative radial solutions when is large

allocation and scheduling of conditional task graphs we propose an original complete and efficient approach to the allocation and scheduling of conditional task graphs ctgs in ctgs nodes represent activities some of them are branches and are labeled with a condition arcs rooted in branch nodes are labeled with condition outcomes and a corresponding probability a task is executed at run time if the condition outcomes that label the arcs in the path to the task hold at schedule execution time this can be captured off line by adopting a stochastic model tasks need for their execution either unary or cumulative resources and some tasks can be executed on alternative resources the solution to the problem is a single assignment of a resource and of a start time to each task so that the allocation and schedule is feasible in each scenario and the expected value of a given objective function is optimized for this problem we need to extend traditional constraint based scheduling techniques in two directions i compute the probability of sets of scenarios in polynomial time in order to get the expected value of the objective function ii define conditional constraints that ensure feasibility in all scenarios we show the application of this framework on problems with objective functions depending either on the allocation of resources to tasks or on the scheduling part also we present the conditional extension to the timetable global constraint experimental results show the effectiveness of the approach on a set of benchmarks taken from the field of embedded system design comparing our solver with a scenario based solver proposed in the literature we show the advantages of our approach both in terms of execution time and solution quality

uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball abstract we study positive radial solutions to u k x f u x e where 0 is a parameter e x r n x r 0 r 0 0 n 2 is the laplacian operator k c r 0 0 satisfies k r 1 r n 0 for r 1 and f c 1 0 r is a concave increasing function satisfying lim s f s s 0 and f 0 0 semipositone we are interested in solutions u such that u 0 as x and satisfy the nonlinear boundary condition u c u u 0 if x r 0 where is the outward normal derivative and c c 0 0 is an increasing function we will establish the uniqueness of positive radial solutions for large values of the parameter

probabilistic set covering with correlations we consider two variants of a probabilistic set covering psc problem the first variant assumes that there is uncertainty regarding whether a selected set can cover an item and the objective is to determine a minimum cost combination of sets so that each item is covered with a prespecified probability the second variant seeks to maximize the minimum probability that a selected set can cover all items to date literature on this problem has focused on the special case in which uncertainties are independent in this paper we formulate deterministic mixed integer programming models for distributionally robust psc problems with correlated uncertainties by exploiting the supermodularity of certain substructures and analyzing their polyhedral properties we develop strong valid inequalities to strengthen the formulations computational results illustrate that our modeling approach can outperform formulations in which correlations are ignored and that our algorithms can significantly reduce overall comput

robust stochastic approximation approach to stochastic programming in this paper we consider optimization problems where the objective function is given in a form of the expectation a basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy the aim of this paper is to compare two computational approaches based on monte carlo sampling techniques namely the stochastic approximation sa and the sample average approximation saa methods both approaches the sa and saa methods have a long history current opinion is that the saa method can efficiently use a specific say linear structure of the considered problem while the sa approach is a crude subgradient method which often performs poorly in practice we intend to demonstrate that a properly modified sa approach can be competitive and even significantly outperform the saa method for a certain class of convex stochastic problems we extend the analysis to the case of convex concave stochastic saddle point problems and present in our opinion highly encouraging results of numerical experiments

an optimal method for stochastic composite optimization this paper considers an important class of convex programming cp problems namely the stochastic composite optimization sco whose objective function is given by the summation of general nonsmooth and smooth stochastic components since sco covers non smooth smooth and stochastic cp as certain special cases a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming note however that the optimization algorithms that can achieve this lower bound had never been developed in this paper we show that the simple mirror descent stochastic approximation method exhibits the best known rate of convergence for solving these problems our major contribution is to introduce the accelerated stochastic approximation ac sa algorithm based on nesterov s optimal method for smooth cp nesterov in doklady an sssr 269 543 547 1983 nesterov in math program 103 127 152 2005 and show that the ac sa algorithm can achieve the aforementioned lower bound on the rate of convergence for sco to the best of our knowledge it is also the first universally optimal algorithm in the literature for solving non smooth smooth and stochastic cp problems we illustrate the significant advantages of the ac sa algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems

provably near optimal sampling based policies for stochastic inventory control models in this paper we consider two fundamental inventory models the single period newsvendor problem and its multiperiod extension but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions under the assumption that the demand distributions are given explicitly these models are well studied and relatively straightforward to solve however in most real life scenarios the true demand distributions are not available or they are too complex to work with thus a sampling driven algorithmic framework is very attractive both in practice and in theory we shall describe how to compute sampling based policies that is policies that are computed based only on observed samples of the demands without any access to or assumptions on the true demand distributions moreover we establish bounds on the number of samples required to guarantee that with high probability the expected cost of the sampling based policies is arbitrarily close i e with arbitrarily small relative error compared to the expected cost of the optimal policies which have full access to the demand distributions the bounds that we develop are general easy to compute and do not depend at all on the specific demand distributions

uniqueness of nonnegative solutions for semipositone problems on exterior domains abstract we consider the problem u k x f u x u 0 if x r 0 u 0 as x where is a positive parameter u div u is the laplacian of u x r n n 2 x r 0 k c 1 r 0 0 is such that lim r k r 0 and f c 1 0 r is a concave function which is sublinear at and f 0 0 we establish the uniqueness of nonnegative radial solutions when is large

exact multiplicity for degenerate two point boundary value problems with p convex nonlinearity the exact number of positive solutions of a degenerate quasilinear two point boundary value problem is investigated for the generalization of earlier results concerning the nondegenerate case with convex nonlinearity suitably defined p convex nonlinearities are considered strictly p convex c2 functions having a non negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval we have uniqueness when f 0 0 and 0 1 or 2 solutions when f 0 0 similarly to the non degenerate case provided that the number of solutions is finite however now there may also occur a continuum of solutions connected to a dead core type phenomenon the proof of our results relies on the shooting method for the characterization of the shape of the time map in contrast to the non degenerate case the shooting method does not determine directly the number of solutions owing to the lack of uniqueness of the corresponding ivp exact conditions on the uniqueness of the ivp and in the case of non uniqueness the number and types of its local solutions are given based on this all the positive solutions of the bvp can be compiled

a simple randomised algorithm for convex optimisation we consider maximising a concave function over a convex set by a simple randomised algorithm the strength of the algorithm is that it requires only approximate function evaluations for the concave function and a weak membership oracle for the convex set under smoothness conditions on the function and the feasible set we show that our algorithm computes a near optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision with high probability as an application to which the features of our algorithm are particularly useful we study two stage stochastic programming problems these problems have the property that evaluation of the objective function is p hard under appropriate assumptions on the models therefore as a tool within our randomised algorithm we devise a fully polynomial randomised approximation scheme for these function evaluations under appropriate assumptions on the models moreover we deal with smoothing the feasible set which in two stage stochastic programming is a polyhedron

the data driven newsvendor problem new bounds and insights consider the newsvendor model but under the assumption that the underlying demand distribution is not known as part of the input instead the only information available is a random independent sample drawn from the demand distribution this paper analyzes the sample average approximation saa approach for the data driven newsvendor problem we obtain a new analytical bound on the probability that the relative regret of the saa solution exceeds a threshold this bound is significantly tighter than existing bounds and it matches the empirical accuracy of the saa solution observed in extensive computational experiments this bound reveals that the demand distribution s weighted mean spread affects the accuracy of the saa heuristic

stochastic integer programming limit theorems and confidence intervals we consider empirical approximations sample average approximations of two stage stochastic mixed integer linear programs and derive central limit theorems for the objectives and optimal values the limit theorems are based on empirical process theory and the functional delta method we also show how these limit theorems can be used to derive confidence intervals for optimal values via resampling methods bootstrap subsampling

stochastic programming approach to optimization under uncertainty in this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems we argue that two stage say linear stochastic programming problems can be solved with a reasonable accuracy by monte carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages we discuss an extension of coherent risk measures to a multistage setting and in particular dynamic programming equations for such problems

models for minimax stochastic linear optimization problems with risk aversion we propose a semidefinite optimization sdp model for the class of minimax two stage stochastic linear optimization problems with risk aversion the distribution of second stage random variables belongs to a set of multivariate distributions with known first and second moments for the minimax stochastic problem with random objective we provide a tight sdp formulation the problem with random right hand side is np hard in general in a special case the problem can be solved in polynomial time explicit constructions of the worst case distributions are provided applications in a productiontransportation problem and a single facility minimax distance problem are provided to demonstrate our approach in our experiments the performance of minimax solutions is close to that of data driven solutions under the multivariate normal distribution and better under extremal distributions the minimax solutions thus guarantee to hedge against these worst possible distributions and provide a natural distribution to stress test stochastic optimization problems under distributional ambiguity

a sequential sampling procedure for stochastic programming we develop a sequential sampling procedure for a class of stochastic programs we assume that a sequence of feasible solutions with an optimal limit point is given as input to our procedure such a sequence can be generated by solving a series of sampling problems with increasing sample size or it can be found by any other viable method our procedure estimates the optimality gap of a candidate solution from this sequence if the point estimate of the optimality gap is sufficiently small according to our termination criterion then we stop otherwise we repeat with the next candidate solution from the sequence under an increased sample size we provide conditions under which this procedure i terminates with probability one and ii terminates with a solution that has a small optimality gap with a prespecified probability

optimization of r d project portfolios under endogenous uncertainty project portfolio management deals with the dynamic selection of research and development r d projects and determination of resource allocations to these projects over a planning period given the uncertainties and resource limitations over the planning period the objective is to maximize the expected total discounted return or the expectation of some other function for all projects over a long time horizon we develop a detailed formal description of this problem and the corresponding decision process and then model it as a multistage stochastic integer program with endogenous uncertainty accounting for this endogeneity we propose an efficient solution approach for the resulting model which involves the development of a formulation technique that is amenable to scenario decomposition the proposed solution algorithm also includes an application of the sample average approximation method where the sample problems are solved through lagrangian relaxation and a new lower bounding heuristic the performance of the overall solution procedure is demonstrated using several implementations of the proposed approach

fixed width sequential stopping rules for a class of stochastic programs monte carlo sampling based methods are frequently used in stochastic programming when exact solution is not possible a critical component of monte carlo sampling based methods is determining when to stop sampling to ensure the desired quality of the solutions in this paper we develop stopping rules for sequential sampling procedures that depend on the width of an optimality gap confidence interval estimator the procedures solve a sequence of sampling approximations with increasing sample size to generate solutions and stop when the width of the confidence interval on the current solution s optimality gap plus an inflation factor fall below a prespecified value epsilon we first present a method that takes the schedule of sample sizes as an input and provide guidelines on the growth of sample sizes then we present a method that increases the sample sizes according to the current estimates of the optimality gap the larger the estimates the larger the increases in sample sizes we provide conditi

retrospective optimization of mixed integer stochastic systems using dynamic simplex linear interpolation abstract we propose a family of retrospective optimization ro algorithms for optimizing stochastic systems with both integer and continuous decision variables the algorithms are continuous search procedures embedded in a ro framework using dynamic simplex interpolation rodsi by decreasing dimensions corresponding to the continuous variables of simplex the retrospective solutions become closer to an optimizer of the objective function we present convergence results of rodsi algorithms for stochastic convex systems numerical results show that a simple implementation of rodsi algorithms significantly outperforms some random search algorithms such as genetic algorithm ga and particle swarm optimization pso

a simulated annealing algorithm with constant temperature for discrete stochastic optimization we present a modification of the simulated annealing algorithm designed for solving discrete stochastic optimization problems like the original simulated annealing algorithm our method has the hill climbing feature so it can find global optimal solutions to discrete stochastic optimization problems with many local solutions however our method differs from the original simulated annealing algorithm in that it uses a constant rather than decreasing temperature we consider two approaches for estimating the optimal solution the first approach uses the number of visits the algorithm makes to the different states divided by a normalizer to estimate the optimal solution the second approach uses the state that has the best average estimated objective function value as estimate of the optimal solution we show that both variants of our method are guaranteed to converge almost surely to the set of global optimal solutions and discuss how our work applies in the discrete deterministic optimization setting we also show how both variants can be applied for solving discrete optimization problems when the objective function values are estimated using either transient or steady state simulation finally we include some encouraging numerical results documenting the behavior of the two variants of our algorithm when applied for solving two versions of a particular discrete stochastic optimization problem and compare their performance with that of other variants of the simulated annealing algorithm designed for solving discrete stochastic optimization problems

multistep stochastic mirror descent for risk averse convex stochastic programs based on extended polyhedral risk measures we consider risk averse convex stochastic programs expressed in terms of extended polyhedral risk measures we derive computable confidence intervals on the optimal value of such stochastic programs using the robust stochastic approximation and the stochastic mirror descent smd algorithms when the objective functions are uniformly convex we also propose a multistep extension of the stochastic mirror descent algorithm and obtain confidence intervals on both the optimal values and optimal solutions numerical simulations show that our confidence intervals are much less conservative and are quicker to compute than previously obtained confidence intervals for smd and that the multistep stochastic mirror descent algorithm can obtain a good approximate solution much quicker than its nonmultistep counterpart

performance guarantees for empirical markov decision processes with applications to multiperiod inventory models we consider markov decision processes with unknown transition probabilities and unknown single period expected cost functions and we study a method for estimating these quantities from historical or simulated data the method requires knowledge of the system equations that govern state transitions as well as the single period cost functions but not the single period expected cost functions the estimation procedure is based upon taking expectations with respect to the empirical distribution functions of such data once the estimates are in place the method computes a policy by solving the obtained empirical markov decision process as if the estimates were correct for mdps that satisfy some conditions we provide explicit easily computed expressions for the probability that the procedure will produce a policy whose true expected cost is within any specified absolute distance of the actual optimal expected cost of the true markov decision process we also provide expressions for the number of historical or simulated data values that is sufficient for the procedure to produce a policy whose true expected cost is with a prescribed probability within a prescribed absolute distance of the actual optimal expected cost of the true markov decision process we apply our results to multiperiod inventory models in addition we provide a specialized analysis of such inventory models that also yields relative rather than absolute accuracy guarantees we make comparisons with related results that have recently appeared and we provide numerical examples

smoothing projected gradient method and its application to stochastic linear complementarity problems a smoothing projected gradient spg method is proposed for the minimization problem on a closed convex set where the objective function is locally lipschitz continuous but nonconvex nondifferentiable we show that any accumulation point generated by the spg method is a stationary point associated with the smoothing function used in the method which is a clarke stationary point in many applications we apply the spg method to the stochastic linear complementarity problem slcp and image restoration problems we study the stationary point defined by the directional derivative and provide necessary and sufficient conditions for a local minimizer of the expected residual minimization erm formulation of slcp preliminary numerical experiments using the spg method for solving randomly generated slcp and image restoration problems of large sizes show that the spg method is promising

stochastic sampling for deterministic structural topology optimization with many load cases density based and ground structure approaches abstract we propose an efficient probabilistic method to solve a fully deterministic problem we present a randomized optimization approach that drastically reduces the enormous computational cost of optimizing designs under many load cases for both continuum and truss topology optimization practical structural designs by deterministic topology optimization typically involve many load cases possibly hundreds or more the optimal design minimizes a possibly weighted average of the compliance under each load case or some other objective this means that in each optimization step a large finite element problem must be solved for each load case leading to an enormous computational effort on the contrary the proposed randomized optimization method with stochastic sampling requires the solution of only a few e g 5 or 6 finite element problems large linear systems per optimization step based on simulated annealing we introduce a damping scheme for the randomized approach through numerical examples in two and three dimensions we demonstrate that the randomization algorithm drastically reduces computational cost to obtain similar final topologies and results e g compliance to those of standard algorithms the results indicate that the damping scheme is effective and leads to rapid convergence of the proposed algorithm

an approximate dynamic programming algorithm for monotone value functions many sequential decision problems can be formulated as markov decision processes mdps where the optimal value function or cost to go function can be shown to satisfy a monotone structure in some or all of its dimensions when the state space becomes large traditional techniques such as the backward dynamic programming algorithm i e backward induction or value iteration may no longer be effective in finding a solution within a reasonable time frame and thus we are forced to consider other approaches such as approximate dynamic programming adp we propose a provably convergent adp algorithm called monotone adp that exploits the monotonicity of the value functions to increase the rate of convergence in this paper we describe a general finite horizon problem setting where the optimal value function is monotone present a convergence proof for monotone adp under various technical assumptions and show numerical results for three application domains optimal stopping energy storage allocation and glycemic control for diabetes patients the empirical results indicate that by taking advantage of monotonicity we can attain high quality solutions within a relatively small number of iterations using up to two orders of magnitude less computation than is needed to compute the optimal solution exactly

sample average approximation of stochastic dominance constrained programs in this paper we study optimization problems with second order stochastic dominance constraints this class of problems allows for the modeling of optimization problems where a risk averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks here we deal with the case of multi variate stochastic dominance under general distributions and nonlinear functions we introduce the concept of mathcal c dominance which generalizes some notions of multi variate dominance found in the literature we apply the sample average approximation saa method to this problem which results in a semi infinite program and study asymptotic convergence of optimal values and optimal solutions as well as the rate of convergence of the feasibility set of the resulting semi infinite program as the sample size goes to infinity we develop a finitely convergent method to find an epsilon optimal solution of the saa problem an important aspect of our contribution is the construction of practical statistical lower and upper bounds for the true optimal objective value we also show that the bounds are asymptotically tight as the sample size goes to infinity

adaptive search algorithms for discrete stochastic optimization a smooth best response approach this paper considers simulation based optimization of the performance of a regime switching stochastic system over a finite set of feasible configurations inspired by the stochastic fictitious play learning rules in game theory we propose an adaptive random search algorithm that uses a smooth best response sampling strategy and tracks the set of global optima yet distributes the search so that most of the effort is spent on simulating the system performance at the global optima the algorithm responds properly to the random unpredictable jumps of the global optimum even when the observations data are temporally correlated as long as a weak law of large numbers holds numerical examples show that the proposed scheme yields faster convergence and superior efficiency for finite sample lengths compared with several existing random search and pure exploration methods in the literature

diffusive logistic equation with constant yield harvesting i steady states we consider a reaction diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth we prove the existence uniqueness and stability of the maximal steady state solutions under certain conditions and we also classify all steady state solutions under more restricted conditions exact global bifurcation diagrams are obtained in the latter case our method is a combination of comparison arguments and bifurcation theory

learning algorithms for separable approximations of discrete stochastic optimization problems we propose the use of sequences of separable piecewise linear approximations for solving nondifferentiable stochastic optimization problems the approximations are constructed adaptively using a combination of stochastic subgradient information and possibly sample information on the objective function itself we prove the convergence of several versions of such methods when the objective function is separable and has integer break points and we illustrate their behavior on numerical examples we then demonstrate the performance on nonseparable problems that arise in the context of two stage stochastic programming problems and demonstrate that these techniques provide near optimal solutions with a very fast rate of convergence compared with other solution techniques

stochastic binary problems with simple penalties for capacity constraints violations this paper studies stochastic programs with first stage binary variables and capacity constraints using simple penalties for capacities violations in particular we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly mathcal n mathcal p hard in general we provide pseudo polynomial algorithms for three special cases of the problem constant weights and capacity uniformly distributed subset sum with gaussian weights and strictly positively distributed random capacity and subset sum with constant weights and arbitrary random capacity we then turn to a branch and cut algorithm based on the outer approximation of the objective function we provide computational results for the stochastic knapsack problem i with gaussian weights and constant capacity and ii with constant weights and capacity uniformly distributed on randomly generated instances inspired by computational results for the knapsack problem

optimal stochastic approximation algorithms for strongly convex stochastic composite optimization i a generic algorithmic framework in this paper we present a generic algorithmic framework namely the accelerated stochastic approximation ac sa algorithm for solving strongly convex stochastic composite optimization sco problems while the classical stochastic approximation algorithms are asymptotically optimal for solving differentiable and strongly convex problems the ac sa algorithm when employed with proper stepsize policies can achieve optimal or nearly optimal rates of convergence for solving different classes of sco problems during a given number of iterations moreover we investigate these ac sa algorithms in more detail such as by establishing the large deviation results associated with the convergence rates and introducing an efficient validation procedure to check the accuracy of the generated solutions

on the stability properties of nonnegative solutions of semilinear problems with convex or concave nonlinearity 497 abstract we investigate the stability of nontrivial nonnegative stationary solutions of semilinear initial boundary value problems with convex or concave nonlinearity in the convex case f 0 0 implies instability in the concave case f 0 0 implies stability we also discuss the necessity of the sign condition on f 0

variable number sample path optimization the sample path method is one of the most important tools in simulation based optimization the basic idea of the method is to approximate the expected simulation output by the average of sample observations with a common random number sequence in this paper we describe a new variant of powell s unconstrained optimization by quadratic approximation uobyqa method which integrates a bayesian variable number sample path vnsp scheme to choose appropriate number of samples at each iteration the statistically accurate scheme determines the number of simulation runs and guarantees the global convergence of the algorithm the vnsp scheme saves a significant amount of simulation operations compared to general purpose fixed number sample path methods we present numerical results based on the new algorithm

positive solution curves of semipositone problems with concave nonlinearities we consider the positive solutions to the semilinear equation where denotes a smooth bounded region in n n 1 and 0 here f 0 is assumed to be monotonically increasing concave and such that f 0 f lim t f t 0 we establish the stability and uniqueness of large positive solutions in terms of f t t when is a ball we determine the exact number of positive solutions for each 0 we also obtain the geometry of the branches of positive solutions completely and establish how they evolve this work extends and complements that of 3 7 where f 0

optimal allocation of simulation experiments in discrete stochastic optimization and approximative algorithms approximate solutions for discrete stochastic optimization problems are often obtained via simulation it is reasonable to complement these solutions by confidence regions for the argmin set we address the question how a certain total number of random draws should be distributed among the set of alternatives two goals are considered the minimization of the costs caused by using a statistical estimate of the true argmin and the minimization of the expected size of the confidence sets we show that an asymptotically optimal sampling strategy in the case of normal errors can be obtained by solving a convex optimization problem to reduce the computational effort we propose a regularization that leads to a simple one step allocation rule

a general framework on the simulation based optimization under fixed computing budget in many real world problems the design space is huge and the estimation of performance measure has to rely on simulation which is time consuming hence to find the optimal design in the design space based on the simulation output is not trivial it is important to have a computing time allocation rule to decide how much effort to spend in sampling the design space how many designs to sample and how long to run for each design alternative within a given computing budget in this paper we propose a framework for making these allocation decisions we use the problem of assemble to order ato systems to demonstrate how this framework can be applied the sample average approximation saa method is chosen as the sampling method used in this application example the numerical results show that this framework provides a good basis for allocation decisions

stochastic combinatorial optimization with controllable risk aversion level most of the recent work on 2 stage stochastic combinatorial optimization problems has focused on minimization of the expected cost or the worst case cost of the solution those two objectives can be viewed as two extreme ways of handling risk in this paper we propose to use a one parameter family of functionals to interpolate between them although such a family has been used in the mathematical finance and stochastic programming literature before its use in the context of approximation algorithms seems new we show that under standard assumptions a broad class of risk adjusted 2 stage stochastic programs can be efficiently treated by the sample average approximation saa method in particular our result shows that it is computationally feasible to incorporate some degree of robustness even when the underlying distribution can only be accessed in a black box fashion we also show that when combined with suitable rounding procedures our result yields new approximation algorithms for many risk adjusted 2 stage stochastic combinatorial optimization problems under the black box setting

confidence based reasoning in stochastic constraint programming in this work we introduce a novel approach based on sampling for finding assignments that are likely to be solutions to stochastic constraint satisfaction problems and constraint optimisation problems our approach reduces the size of the original problem being analysed by solving this reduced problem with a given confidence probability we obtain assignments that satisfy the chance constraints in the original model within prescribed error tolerance thresholds to achieve this we blend concepts from stochastic constraint programming and statistics we discuss both exact and approximate variants of our method the framework we introduce can be immediately employed in concert with existing approaches for solving stochastic constraint programs a thorough computational study on a number of stochastic combinatorial optimisation problems demonstrates the effectiveness of our approach

sample average approximation method for compound stochastic optimization problems the paper studies stochastic optimization programming problems with compound functions containing expectations and extreme values of other random functions as arguments compound functions arise in various applications a typical example is a variance function of nonlinear outcomes other examples include stochastic minimax problems econometric models with latent variables and multilevel and multicriteria stochastic optimization problems as a solution technique a sample average approximation saa method also known as statistical or empirical sample mean method is used the method consists in approximation of all expectation functions by their empirical means and solving the resulting approximate deterministic optimization problems in stochastic optimization this method is widely used for optimization of standard expectation functions under constraints in this paper saa method is extended to general compound stochastic optimization problems the conditions for convergence in mean almost sure

unbounded connected component of the positive solution set of nonlinear operator equation and its applications abstract in this paper by using global bifurcation theories we obtain some results for structure of positive solution set of some nonlinear equations with parameters as a result we obtain some existence results for positive solutions of the nonlinear operator equation the main result can be applied to various of differential boundary value problems to obtain the existence results for positive solutions without the assumption that the nonlinearities are positone

stochastic global optimization using tangent minorants for lipschitz functions abstract this paper deals with global stochastic optimization where the decision variable belongs to a compact subset x of r the objective function is the mathematical expectation of a partial bivariate lipschitz function f x depending on a decision variable x and a random variable whose probability distribution depends on x in the first part of the present paper we propose a branch and bound algorithm based on tangent minorants that provides a global minimum in the second part we consider the case where the function f is discontinuous we show the manner we correct that discontinuity without modifying the global minimum of e f x we also illustrate how to extend this framework to multidimensional stochastic optimization problems by using the alienor method then we validate the proposed method by applying it to some test functions and compare it to known algorithms

solving stochastic programs with integer recourse by enumeration a framework using gro bner basis reductions in this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right hand side vector using grobner basis methods from computational algebra to solve the numerous second stage integer programs using structural properties of the expected integer recourse function we prove that under mild conditions an optimal solution is contained in a finite set furthermore we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed

integer programming approaches for appointment scheduling with random no shows and service durations we consider a single server scheduling problem given a fixed sequence of appointment arrivals with random no shows and service durations the probability distribution of the uncertain parameters is assumed to be ambiguous and only the support and first moments are known we formulate a class of distributionally robust dr optimization models that incorporate the worst case expectation conditional value at risk penalty cost of appointment waiting server idleness and overtime into the objective or constraints our models flexibly adapt to different prior beliefs of no show uncertainty we obtain exact mixed integer nonlinear programming reformulations and derive valid inequalities to strengthen the reformulations that are solved by decomposition algorithms in particular we derive convex hulls for special cases of no show beliefs yielding polynomial sized linear programming models for the least and the most conservative supports of no shows we test various instances to demonstrate the computational e

a sequential algorithm for solving nonlinear optimization problems with chance constraints an algorithm is presented for solving nonlinear optimization problems with chance constraints i e those in which a constraint involving an uncertain parameter must be satisfied with at least a minimum probability in particular the algorithm is designed to solve cardinality constrained nonlinear optimization problems that arise in sample average approximations of chance constrained problems as well as in other applications in which it is only desired to enforce a minimum number of constraints the algorithm employs a novel exact penalty function which is minimized sequentially by solving quadratic optimization subproblems with linear cardinality constraints properties of minimizers of the penalty function in relation to minimizers of the corresponding nonlinear optimization problem are presented and convergence of the proposed algorithm to stationarity with respect to the penalty function is proved the effectiveness of the algorithm is demonstrated through numerical experiments with a nonlinear ca

adaptive adversarial multi armed bandit approach to two person zero sum markov games this technical note presents a recursive sampling based algorithm for finite horizon two person zero sum markov games mgs based on the exp3 algorithm developed by auer et al for adaptive adversarial multi armed bandit problems we provide a finite iteration bound to the equilibrium value of the induced sample average approximation game of a given mg and prove asymptotic convergence to the equilibrium value of the given mg the time and space complexities of the algorithm are independent of the state space of the game

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 rates of convergence for stochastic optimization problems under non independent and identically distributed sampling in this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size this is a well studied problem in case the samples are independent and identically distributed i e when standard monte carlo simulation is used here we study the case where that assumption is dropped broadly speaking our results show that under appropriate assumptions the rates of convergence for pointwise estimators under a sampling scheme carry over to the optimization case in the sense that convergence of approximating optimal solutions and optimal values to their true counterparts has the same rates as in pointwise estimation we apply our results to two well established sampling schemes namely latin hypercube sampling and randomized quasi monte carlo qmc the novelty of our work arises from the fact that while there has been some work on the use of variance reduction techniques and qmc methods in stochastic optimization none of the existing work to the best of our knowledge has provided a theoretical study on the effect of these techniques on rates of convergence for the optimization problem we present numerical results for some two stage stochastic programs from the literature to illustrate the discussed ideas

extreme canonical correlations and high dimensional cointegration analysis abstract we prove that the extreme squared sample canonical correlations between a random walk and its own innovations almost surely converge to the upper and lower boundaries of the support of the wachter distribution when the sample size and the dimensionality go to infinity proportionally this result is used to derive previously unknown analytic expressions for the bartlett type correction coefficients for johansen s trace and maximum eigenvalue tests in a high dimensional var 1 an analysis of cointegration among a large number of log exchange rates illustrates the usefulness of our theoretical results

optimal stochastic approximation algorithms for strongly convex stochastic composite optimization ii shrinking procedures and optimal algorithms in this paper we study new stochastic approximation sa type algorithms namely the accelerated sa ac sa for solving strongly convex stochastic composite optimization sco problems specifically by introducing a domain shrinking procedure we significantly improve the large deviation results associated with the convergence rate of a nearly optimal ac sa algorithm presented by ghadimi and lan in siam j optim 22 2012 pp 1469 1492 moreover we introduce a multistage ac sa algorithm which possesses an optimal rate of convergence for solving strongly convex sco problems in terms of the dependence on not only the target accuracy but also a number of problem parameters and the selection of initial points to the best of our knowledge this is the first time that such an optimal method has been presented in the literature from our computational results these ac sa algorithms can substantially outperform the classical sa and some other sa type algorithms for solving certain classes of strongly

assessing solution quality in stochastic programs determining whether a solution is of high quality optimal or near optimal is fundamental in optimization theory and algorithms in this paper we develop monte carlo sampling based procedures for assessing solution quality in stochastic programs quality is defined via the optimality gap and our procedures output is a confidence interval on this gap we review a multiple replications procedure that requires solution of say 30 optimization problems and then we present a result that justifies a computationally simplified single replication procedure that only requires solving one optimization problem even though the single replication procedure is computationally significantly less demanding the resulting confidence interval might have low coverage probability for small sample sizes for some problems we provide variants of this procedure that require two replications instead of one and that perform better empirically we present computational results for a newsvendor problem and for two stage stochastic linear programs from the literature we also discuss when the procedures perform well and when they fail and we propose using optimal solutions to strengthen the performance of our procedures

sleeping experts and bandits approach to constrained markov decision processes this communique presents simple simulation based algorithms for obtaining an approximately optimal policy in a given finite set in large finite constrained markov decision processes the algorithms are adapted from playing strategies for sleeping experts and bandits problem and their computational complexities are independent of state and action space sizes if the given policy set is relatively small we establish convergence of their expected performances to the value of an optimal policy and convergence rates and also almost sure convergence to an optimal policy with an exponential rate for the algorithm adapted within the context of sleeping experts

an asymptotically efficient simulation based algorithm for finite horizon stochastic dynamic programming we present a simulation based algorithm called simulated annealing multiplicative weights samw for solving large finite horizon stochastic dynamic programming problems at each iteration of the algorithm a probability distribution over candidate policies is updated by a simple multiplicative weight rule and with proper annealing of a control parameter the generated sequence of distributions converges to a distribution concentrated only on the best policies the algorithm is asymptotically efficient in the sense that for the goal of estimating the value of an optimal policy a provably convergent finite time upper bound for the sample mean is obtained

on optimal allocation of indivisibles under uncertainty the optimal allocation of indivisible resources is formalized as a stochastic optimization problem involving discrete decision variables a general stochastic search procedure is proposed which develops the concept of the branch and bound method the main idea is to process large collections of possible solutions and to devote more attention to the most promising groups by gathering more information to reduce the uncertainty and by narrowing the search area the optimal solution can be found with probability one special techniques for calculating stochastic lower and upper bounds are discussed the results are illustrated by a computational experiment

divide to conquer decomposition methods for energy optimization modern electricity systems provide a plethora of challenging issues in optimization the increasing penetration of low carbon renewable sources of energy introduces uncertainty in problems traditionally modeled in a deterministic setting the liberalization of the electricity sector brought the need of designing sound markets ensuring capacity investments while properly reflecting strategic interactions in all these problems hedging risk possibly in a dynamic manner is also a concern the fact of representing uncertainty and or competition of different companies in a multi settlement power market considerably increases the number of variables and constraints for this reason usually a trade off needs to be found between modeling and numerical tractability the more details are brought into the model the harder becomes the optimization problem for structured optimization and generalized equilibrium problems we explore some variants of solution methods based on lagrangian relaxation and on benders decomposition throughout we keep as a leading thread the actual practical value of such techniques in terms of their efficiency to solve energy related problems

two stage quadratic integer programs with stochastic right hand sides we consider two stage quadratic integer programs with stochastic right hand sides and present an equivalent reformulation using value functions we propose a two phase solution approach the first phase constructs value functions of quadratic integer programs in both stages the second phase solves the reformulation using a global branch and bound algorithm or a level set approach we derive some basic properties of value functions of quadratic integer programs and utilize them in our algorithms we show that our approach can solve instances whose extensive forms are hundreds of orders of magnitude larger than the largest quadratic integer programming instances solved in the literature

profit estimation error in the newsvendor model under a parametric demand distribution we consider the newsvendor model in which uncertain demand is assumed to follow a probabilistic distribution with known functional form but unknown parameters these parameters are estimated unbia

a note on appointment scheduling with piecewise linear cost functions we consider the problem of determining the optimal schedules for a given sequence of jobs on a single processor the objective is to minimize the expected total cost incurred by job waiting and processor idling where the job processing times are random variables it is known in the prior literature that if the processing times are integers and the costs are linear functions satisfying a mild condition then the problem can be solved in a polynomial number of expected cost evaluations in this work we extend the result to piecewise linear cost functions which include many useful objective functions in practice our analysis explores the hidden dual network flow structure of the appointment scheduling problem and thus greatly simplifies that of prior work we also find the number of samples needed to compute a near optimal solution when only some independent samples of processing times are known

existence of positive radial solutions for a semipositone elliptic equation abstract in this paper we study the existence of positive radial solutions for a semipositone elliptic equation u f u in u 0 on where is a ball or an annulus in r n with n 2 0 is a parameter and f is a continuous function satisfying the semipositone condition f 0 0 we give a weak and general sufficient condition on f for the existence of positive radial solutions for 0 large we show that the linearized operator at each solution has the nonnegative first eigenvalue these solutions are obtained as minimizers of a lagrangian functional

robust sample average approximation sample average approximation saa is a widely popular approach to data driven decision making under uncertainty under mild assumptions saa is both tractable and enjoys strong asymptotic performance guarantees similar guarantees however do not typically hold in finite samples in this paper we propose a modification of saa which we term robust saa which retains saa s tractability and asymptotic properties and additionally enjoys strong finite sample performance guarantees the key to our method is linking saa distributionally robust optimization and hypothesis testing of goodness of fit beyond robust saa this connection provides a unified perspective enabling us to characterize the finite sample and asymptotic guarantees of various other data driven procedures that are based upon distributionally robust optimization this analysis provides insight into the practical performance of these various methods in real applications we present examples from inventory management and portfolio allocation and demonstrate numerically that our approach outperforms other data driven approaches in these applications

stochastic root finding and efficient estimation of convex risk measures reliable risk measurement is a key problem for financial institutions and regulatory authorities the current industry standard value at risk has several deficiencies improved risk measures have been suggested and analyzed in the recent literature but their computational implementation has largely been neglected so far we propose and investigate stochastic approximation algorithms for the convex risk measure utility based shortfall risk our approach combines stochastic root finding schemes with importance sampling we prove that the resulting shortfall risk estimators are consistent and asymptotically normal and provide formulas for confidence intervals the performance of the proposed algorithms is tested numerically we finally apply our techniques to the normal copula model which is also known as the industry model creditmetrics this provides guidance for future implementations in practice

the static stochastic knapsack problem with normally distributed item sizes this paper develops exact and heuristic algorithms for a stochastic knapsack problem where items with random sizes may be assigned to a knapsack an item s value is given by the realization of the product of a random unit revenue and the random item size when the realization of the sum of selected item sizes exceeds the knapsack capacity a penalty cost is incurred for each unit of overflow while our model allows for a salvage value for each unit of capacity that remains unused we seek to maximize the expected net profit resulting from the assignment of items to the knapsack although the capacity is fixed in our core model we show that problems with random capacity as well as problems in which capacity is a decision variable subject to unit costs fall within this class of problems as well we focus on the case where item sizes are independent and normally distributed random variables and provide an exact solution method for a continuous relaxation of the problem we show that an optimal solution to this relaxation exists containing no more than two fractionally selected items and develop a customized branch and bound algorithm for obtaining an optimal binary solution in addition we present an efficient heuristic solution method based on our algorithm for solving the relaxation and empirically show that it provides high quality solutions

conditioning of convex piecewise linear stochastic programs in this paper we consider stochastic programming problems where the objective function is given as an expected value of a convex piecewise linear random function with an optimal solution of such a problem we associate a condition number which characterizes well or ill conditioning of the problem using theory of large deviations we show that the sample size needed to calculate the optimal solution of such problem with a given probability is approximately proportional to the condition number

a branch and bound method for stochastic global optimization a stochastic branch and bound method for solving stochastic global optimization problems is proposed as in the deterministic case the feasible set is partitioned into compact subsets to guide the partitioning process the method uses stochastic upper and lower estimates of the optimal value of the objective function in each subset convergence of the method is proved and random accuracy estimates derived methods for constructing stochastic upper and lower bounds are discussed the theoretical considerations are illustrated with an example of a facility location problem

on choosing parameters in retrospective approximation algorithms for stochastic root finding and simulation optimization the stochastic root finding problem is that of finding a zero of a vector valued function known only through a stochastic simulation the simulation optimization problem is that of locating a real valued function s minimum again with only a stochastic simulation that generates function estimates retrospective approximation ra is a sample path technique for solving such problems where the solution to the underlying problem is approached via solutions to a sequence of approximate deterministic problems each of which is generated using a specified sample size and solved to a specified error tolerance our primary focus in this paper is providing guidance on choosing the sequence of sample sizes and error tolerances in ra algorithms we first present an overview of the conditions that guarantee the correct convergence of ra s iterates then we characterize a class of error tolerance and sample size sequences that are superior to others in a certain precisely defined sense we also identify and recommend members of this class and provide a numerical example illustrating the key results

positive solutions for infinite semipositone problems with falling zeros abstract we consider the positive solutions to the singular problem p u a u f u c u in u 0 on where 0 1 a 0 and c 0 are constants is a bounded domain with smooth boundary and f 0 r is a continuous function we assume that there exist m 0 a 0 p 1 such that a u m f u a u p for all u 0 a simple example of f satisfying these assumptions is f u u p for any p 1 we use the method of sub supersolutions to prove the existence of a positive solution of p when a 2 1 1 and c is small here 1 is the first eigenvalue of operator with dirichlet boundary conditions we also extend our result to classes of infinite semipositone systems