Class Of Stochastic Optimization Problems Research Articles

A Symmetric Linearized Alternating Direction Method of Multipliers for a Class of Stochastic Optimization Problems

A Symmetric Linearized Alternating Direction Method of Multipliers for a Class of Stochastic Optimization Problems

A Simulation Optimization Approach for the Appointment Scheduling Problem with Decision-Dependent Uncertainties

The appointment scheduling problem (ASP) studies how to manage patient arrivals to a healthcare system to improve system performance. An important challenge occurs when some patients may not show up for an appointment. Although the ASP is well studied in the literature, the vast majority of the existing work does not consider the well-observed phenomenon that patient no-show is influenced by the appointment time, the usual decision variable in the ASP. This paper studies the ASP with random service time (exogenous uncertainty) with known distribution and patient decision-dependent no-show behavior (endogenous uncertainty). This problem belongs to the class of stochastic optimization with decision-dependent uncertainties. Such problems are notoriously difficult as they are typically nonconvex. We propose a stochastic projected gradient path (SPGP) method to solve the problem, which requires the development of a gradient estimator of the objective function—a nontrivial task, as the literature on gradient-based optimization algorithms for problems with decision-dependent uncertainty is very scarce and unsuitable for our model. Our method can solve the ASP problem under arbitrarily smooth show-up probability functions. We present solutions under different patterns of no-show behavior and demonstrate that breaking the assumption of constant show-up probability substantially changes the scheduling solutions. We conduct numerical experiments in a variety of settings to compare our results with those obtained with a distributionally robust optimization method developed in the literature. The cost reduction obtained with our method, which we call the value of distribution information, can be interpreted as how much the system performance can be improved by knowing the distribution of the service times, compared to not knowing it. We observe that the value of distribution information is up to 31% of the baseline cost, and that such value is higher when the system is crowded or/and the waiting time cost is relatively high. Summary of Contribution: This paper studies an appointment scheduling problem under time-dependent patient no-show behavior, a situation commonly observed in practice. The problem belongs to the class of stochastic optimization problems with decision-dependent uncertainties in the operations research literature. Such problems are notoriously difficult to solve as a result of the lack of convexity, and the present case requires different techniques because of the presence of continuous distributions for the service times. A stochastic projected gradient path method, which includes the development of specialized techniques to estimate the gradient of the objective function, is proposed to solve the problem. For problems with a similar structure, the algorithm can be applied once the gradient estimator of the objective function is obtained. Extensive numerical studies are presented to demonstrate the quality of the solutions, the importance of modeling time-dependent no-shows in appointment scheduling, and the value of using distribution information about the service times.

Models and algorithms for decomposition problems

This thesis deals with the decomposition both as a solution method and as a problem itself. A decomposition approach can be very effective for mathematical problems presenting a specific structure in which the associated matrix of coefficients is sparse and it is diagonalizable in blocks. But, this kind of structure may not be evident from the most natural formulation of the problem. Thus, its coefficient matrix may be preprocessed by solving a structure detection problem in order to understand if a decomposition method can successfully be applied. So, this thesis deals with the k-Vertex Cut problem, that is the problem of finding the minimum subset of nodes whose removal disconnects a graph into at least k components, and it models relevant applications in matrix decomposition for solving systems of equations by parallel computing. The capacitated k-Vertex Separator problem, instead, asks to find a subset of vertices of minimum cardinality the deletion of which disconnects a given graph in at most k shores and the size of each shore must not be larger than a given capacity value. Also this problem is of great importance for matrix decomposition algorithms. This thesis also addresses the Chance-Constrained Mathematical Program that represents a significant example in which decomposition techniques can be successfully applied. This is a class of stochastic optimization problems in which the feasible region depends on the realization of a random variable and the solution must optimize a given objective function while belonging to the feasible region with a probability that must be above a given value. In this thesis, a decomposition approach for this problem is introduced. The thesis also addresses the Fractional Knapsack Problem with Penalties, a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity.

Sample Complexity of Sample Average Approximation for Conditional Stochastic Optimization

In this paper, we study a class of stochastic optimization problems, referred to as the conditional stochastic optimization (CSO), in the form of $\min_{x \in \mathcal{X}} \mathbb{E}_{\xi}f_\xi({\mathbb{E}_{\eta|\xi}[g_\eta(x, \xi)]})$, which finds a wide spectrum of applications including portfolio selection, reinforcement learning, robust learning, and causal inference. Assuming availability of samples from the distribution $\mathbb{P}(\xi)$ and samples from the conditional distribution $\mathbb{P}(\eta|\xi)$, we establish the sample complexity of the sample average approximation (SAA) for CSO, under a variety of structural assumptions, such as Lipschitz continuity, smoothness, and error bound conditions. We show that the total sample complexity improves from $\mathcal{O}(d/\epsilon^4)$ to $\mathcal{O}(d/\epsilon^3)$ when assuming smoothness of the outer function, and further to $\mathcal{O}(1/\epsilon^2)$ when the empirical function satisfies the quadratic growth condition. We also establish the sample complexity of a modified SAA when $\xi$ and $\eta$ are independent. Several numerical experiments further support our theoretical findings.

On the Convergence of Mirror Descent beyond Stochastic Convex Programming

In this paper, we examine the convergence of mirror descent in a class of stochastic optimization problems that are not necessarily convex (or even quasi-convex) and which we call variationally coh...

Revisiting Stochastic Loss Networks: Structures and Approximations

We consider fundamental properties of stochastic loss networks, seeking to improve on the so-called Erlang fixed-point approximation. We propose a family of mathematical approximations for estimating the stationary loss probabilities and show that they always converge exponentially fast, provide asymptotically exact results, and yield greater accuracy than the Erlang fixed-point approximation. We further derive structural properties of the inverse of the classical Erlang loss function that characterize the region of capacities that ensures a workload is served within a set of loss probabilities. We then exploit these results to efficiently solve a general class of stochastic optimization problems involving loss networks. Computational experiments investigate various issues of both theoretical and practical interest, and demonstrate the benefits of our approach.

Weakly Coupled Dynamic Program: Information and Lagrangian Relaxations

"Weakly coupled dynamic program" describes a broad class of stochastic optimization problems in which multiple controlled stochastic processes evolve independently but subject to a set of linking constraints imposed on the controls. One feature of the weakly coupled dynamic program is that it decouples into lower-dimensional dynamic programs by dualizing the linking constraint via the Lagrangian relaxation, which also yields a bound on the optimal value of the original dynamic program. Together with the Lagrangian bound, we utilize the information relaxation approach that relaxes the non-anticipative constraint on the controls to obtain a tighter dual bound. We also investigate other combinations of the relaxations and place the resulting bounds in order. To tackle large-scale problems, we further propose a computationally tractable method based on information relaxation, and provide insightful interpretation and performance guarantee. We implement our method and demonstrate its use through two numerical examples.

Distributed Stochastic Optimization via Matrix Exponential Learning

In this paper, we investigate a distributed learning scheme for a broad class of stochastic optimization problems and games that arise in signal processing and wireless communications. The proposed algorithm relies on the method of matrix exponential learning (MXL) and only requires locally computable gradient observations that are possibly imperfect and/or obsolete. To analyze it, we introduce the notion of a stable Nash equilibrium and we show that the algorithm is globally convergent to such equilibria - or locally convergent when an equilibrium is only locally stable. We also derive an explicit linear bound for the algorithm's convergence speed, which remains valid under measurement errors and uncertainty of arbitrarily high variance. To validate our theoretical analysis, we test the algorithm in realistic multi-carrier/multiple-antenna wireless scenarios where several users seek to maximize their energy efficiency. Our results show that learning allows users to attain a net increase between 100% and 500% in energy efficiency, even under very high uncertainty.

Cut generation for optimization problems with multivariate risk constraints

We consider a class of stochastic optimization problems that features benchmarking preference relations among random vectors representing multiple random performance measures (criteria) of interest. Given a benchmark random performance vector, preference relations are incorporated into the model as constraints, which require the decision-based random vector to be preferred to the benchmark according to a relation based on multivariate conditional value-at-risk (CVaR) or second-order stochastic dominance (SSD). We develop alternative mixed-integer programming formulations and solution methods for cut generation problems arising in optimization under such multivariate risk constraints. The cut generation problems for CVaR- and SSD-based models involve the epigraphs of two distinct piecewise linear concave functions, which we refer to as reverse concave sets. We give the complete linear description of the linearization polytopes of these two non-convex substructures. We present computational results that show the effectiveness of our proposed models and methods.

Online Sequential Optimization with Biased Gradients: Theory and Applications to Censored Demand

In this paper, we study a class of stochastic optimization problems, where although the objective functions may not be convex, they satisfy a generalization of convexity called the sequentially convex property. We focus on a setting where the distribution of the underlying uncertainty is unknown and the manager must make a decision in real time based on historical data. Because sequentially convex functions are not necessarily convex, they pose difficulties in applying standard adaptive methods for convex optimization. We propose a nonparametric algorithm based on a gradient descent method and show that the T-season average expected cost differs from the minimum cost by at most [Formula: see text]. Our analysis is based on a careful quantification of the bias that is inherent in gradient estimation because of the adaptive nature of the problem. We demonstrate the usefulness of the concept of sequential convexity by applying it to three canonical problems in inventory control, capacity allocation, and the lifetime buy decision, under the assumption that the manager does not know the demand distributions and has access only to historical sales (censored demand) data.

The Distribution Function of Airport Taxi-Out Times and Selected Applications

Except in adverse weather conditions, congestion at large airport hubs appears to be predictable. This paper attempts to translate this predictability into a distribution of taxi-out times, a key component of airport congestion. When scheduled flights are chosen to define the dataset, taxi-out times follow a uniform distribution. This is not only the simplest distribution that inferences can be based on, but also a distribution that can be estimated by simple linear regression leading to very accurate forecasts. But above all, it is an invertible distribution function that can help solve a large class of stochastic optimization problems.

Max Weight Learning Algorithms for Scheduling in Unknown Environments

We consider a discrete time queueing system where a controller makes a 2-stage decision every slot. The decision at the first stage reveals a hidden source of randomness with a control-dependent (but unknown) probability distribution. The decision at the second stage generates an attribute vector that depends on this revealed randomness. The goal is to stabilize all queues and optimize a utility function of time average attributes, subject to an additional set of time average constraints. This setting fits a wide class of stochastic optimization problems, including multi-user wireless scheduling with dynamic channel measurement decisions, and wireless multi-hop routing with multi-receiver diversity and opportunistic routing decisions. We develop a simple max-weight algorithm that learns efficient behavior by averaging functionals of previous outcomes.

A class of stochastic optimization problems with application to selective data editing

We present a class of stochastic optimization problems with constraints expressed in terms of expectation and with partial knowledge of the outcome in advance to the decision. The constraints imply that the problem cannot be reduced to a deterministic one. Since the knowledge of the outcome is relevant to the decision, it is necessary to seek the solution in a space of random variables. We prove that under convexity conditions, a duality method can be used to solve the problem. An application to statistical data editing is also presented. The search for a good selective editing strategy is stated as an optimization problem in which the objective is to minimize the expected workload with the constraint that the expected error of the aggregates computed with the edited data is below a certain constant. We present the results of real data experimentation and a comparison with a well-known method.

Entropic Value-at-Risk: A New Coherent Risk Measure

This paper introduces the concept of entropic value-at-risk (EVaR), a new coherent risk measure that corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the value-at-risk (VaR) as well as the conditional value-at-risk (CVaR). We show that a broad class of stochastic optimization problems that are computationally intractable with the CVaR is efficiently solvable when the EVaR is incorporated. We also prove that if two distributions have the same EVaR at all confidence levels, then they are identical at all points. The dual representation of the EVaR is closely related to the Kullback-Leibler divergence, also known as the relative entropy. Inspired by this dual representation, we define a large class of coherent risk measures, called g-entropic risk measures. The new class includes both the CVaR and the EVaR.

An Elementary Approach to Optimal Stopping Problems for AR(1) Sequences

Optimal stopping problems form a class of stochastic optimization problems that has a wide range of applications in sequential statistics and mathematical finance. Here we consider a general optimal stopping problem with discounting for autoregressive processes. Our strategy for a solution consists of two steps: First we give elementary conditions to ensure that an optimal stopping time is of threshold type. Then the resulting one-dimensional problem of finding the optimal threshold is to be solved explicitly. The second step is carried out for the case of exponentially distributed innovations.

Decoupled approach to multidisciplinary design optimization under uncertainty

We propose solution methods for multidisciplinary design optimization (MDO) under uncertainty. This is a class of stochastic optimization problems that engineers are often faced with in a realistic design process of complex systems. Our approach integrates solution methods for reliability-based design optimization (RBDO) with solution methods for deterministic MDO problems. The integration is enabled by the use of a deterministic equivalent formulation and the first order Taylor’s approximation in these RBDO methods. We discuss three specific combinations: the RBDO methods with the multidisciplinary feasibility method, the all-at-once method, and the individual disciplinary feasibility method. Numerical examples are provided to demonstrate the procedure.

On the undecidability of probabilistic planning and related stochastic optimization problems

Automated planning, the problem of how an agent achieves a goal given a repertoire of actions, is one of the foundational and most widely studied problems in the AI literature. The original formulation of the problem makes strong assumptions regarding the agent's knowledge and control over the world, namely that its information is complete and correct, and that the results of its actions are deterministic and known. Recent research in planning under uncertainty has endeavored to relax these assumptions, providing formal and computation models wherein the agent has incomplete or noisy information about the world and has noisy sensors and effectors. This research has mainly taken one of two approaches: extend the classical planning paradigm to a semantics that admits uncertainty, or adopt another framework for approaching the problem, most commonly the Markov Decision Process (MDP) model. This paper presents a complexity analysis of planning under uncertainty. It begins with the “probabilistic classical planning” problem, showing that problem to be formally undecidable. This fundamental result is then applied to a broad class of stochastic optimization problems, in brief any problem statement where the agent (a) operates over an infinite or indefinite time horizon, and (b) has available only probabilistic information about the system's state. Undecidability is established for policy-existence problems for partially observable infinite-horizon Markov decision processes under discounted and undiscounted total reward models, average-reward models, and state-avoidance models. The results also apply to corresponding approximation problems with undiscounted objective functions. The paper answers a significant open question raised by Papadimitriou and Tsitsiklis [Math. Oper. Res. 12 (3) (1987) 441–450] about the complexity of infinite horizon POMDPs.

A class of stochastic optimization problems with one quadratic & several linear objective functions and extended portfolio selection model

In this paper a class of stochastic multiple-objective programming problems with one quadratic, several linear objective functions and linear constraints has been introduced. The former model is transformed into a deterministic multiple-objective nonlinear programming model by means of the introduction of random variables’ expectation. The reference direction approach is used to deal with linear objectives and results in a linear parametric optimization formula with a single linear objective function. This objective function is combined with the quadratic function using the weighted sums. The quadratic problem is transformed into a linear (parametric) complementary problem, the basic formula for the proposed approach. The sufficient and necessary conditions for (properly, weakly) efficient solutions and some construction characteristics of (weakly) efficient solution sets are obtained. An interactive algorithm is proposed based on reference direction and weighted sums. Varying the parameter vector on the right-hand side of the model, the DM can freely search the efficient frontier with the model. An extended portfolio selection model is formed when liquidity is considered as another objective to be optimized besides expectation and risk. The interactive approach is illustrated with a practical example.

Performance optimization of multidisciplinary mechanical systems subject to uncertainties

This paper presents a methodology to solve a new class of stochastic optimization problems for multidisciplinary systems (multidisciplinary stochastic optimization or MSO) wherein the objective is to maximize system mechanical performance (e.g. aerodynamic efficiency) while satisfying reliability-based constraints (e.g. structural safety). Multidisciplinary problems require a different solution approach than those solved in earlier research in reliability-based structural optimization (single discipline) wherein the goal is usually to minimize weight (or cost) for a structural configuration subject to a limiting probability of failure or to minimize probability of failure subject to a limiting weight (or cost). For the problems solved herein, the objective is to maximize performance over the range of operating conditions, while satisfying constraints that ensure safe and reliable operation. Because the objective is performance based and because the constraints are reliability based, the random variables used in the objective must model variability in operating conditions, while the random variables used in the constraints must model uncertainty in extreme values (to ensure safety). Thus, the problem must be formulated to treat these two different types of variables at the same time, including the case when the same physical quantity (e.g. a particular load) appears in both the objective function and the constraints. In addition, the problem must be formulated to treat multiple load cases, which can again require modeling the same physical quantity with different random variables. Deterministic multidisciplinary optimization (MDO) problems have advanced to the stage where they are now commonly formulated with multiple load cases and multiple disciplines governing the objective and constraints. This advancement has enabled MDO to solve more realistic problems of much more practical interest. The formulation used herein solves stochastic optimization problems that are posed in this same way, enabling similar practical benefits but, in addition, producing optimum designs that are more robust than the deterministic optimum designs (since uncertainties are accounted for during the optimization process). The methodology has been implemented in the form of a baseline MSO shell that executes on both a massively parallel computer and a network of workstations. The MSO shell is demonstrated herein by a stochastic shape optimization of an axial compressor blade involving fully coupled aero-structural analysis.

On nonsmooth and discontinuous problems of stochastic systems optimization

A class of stochastic optimization problems is analyzed that cannot be solved by deterministic and standard stochastic approximation methods. We consider risk-control problems, optimization of stochastic networks and discrete event systems, screening irreversible changes, and pollution control. The results of Ermoliev et al. are extended to the case of stochastic systems and general constraints. It is shown that the concept of stochastic mollifier gradient leads to easily implementable computational procedures for systems with Lipschitz and discontinuous objective functions. New optimality conditions are formulated for designing stochastic search procedures for constrained optimization of discontinuous systems.