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We present a problem data set for stochastic programming, and associated real world applications. The problem descriptions were collected from the literature, with emphasis on variety of problem structure and application. Each problem has a short description, mathematical problem statement, and notational reconciliation to a standard problem format. In addition, most problems have one or more corresponding data files in SMPS1 format.

This research investigates a special class of stochastic linear programs known as two-stage stochastic linear programming with relatively complete and fixed recourse. These models characterize a two-phase process where the first- stage decision (itself subject to a separate set of first-stage linear constraints) allocates a set of resources to the second-stage linear program prior to the realization of random variables affecting second-stage resource availability. Since the second-stage decision deterministically follows both first-stage allocation and random variable realization, the first-stage variables constitute the only true decision. The expected cost of the two-stage recourse problem is also a piecewise convex function of the first-stage decision variables, thus allowing a global optimal solution that minimizes the total expected cost.

The general linear programming problem is considered in which the coefficients of the objective function to be maximized are assumed to be random variables with a known multinormal distribution. Three deterministic reformulations involve maximizing the expected value, the alpha-fractile (alpha fixed, 0 alpha 1/2), and the probability of exceeding a predetermined level of payoff, respectively. In this paper the author's previous work on 'bi- criterion programs' is applied to derive an algorithm for routinely and efficiently solving the second and third reformulations. A by-product of the calculations in each case is the tradeoff-curve between the criterion being maximized and expected payoff. The intimate relationships between all three reformulations are illuminated.

The general linear programming problem is considered in which the coefficients of the objective function to be maximized are assumed to be random variables with a known multinormal distribution. Three deterministic reformulations involve maximizing the expected value, the alpha-fractile (alpha fixed, 0 alpha 1/2), and the probability of exceeding a predetermined level of payoff, respectively. In this paper the author's previous work on 'bi- criterion programs' is applied to derive an algorithm for routinely and efficiently solving the second and third reformulations. A by-product of the calculations in each case is the tradeoff-curve between the criterion being maximized and expected payoff. The intimate relationships between all three reformulations are illuminated.

This document studies the dual problem corresponding to a linear program in which the stochastic objective function is replaced by its expected utility, and discusses its relevance as a penalty method to a stochastically constrained dual linear program.

This document studies the dual problem corresponding to a linear program in which the stochastic objective function is replaced by its expected utility, and discusses its relevance as a penalty method to a stochastically constrained dual linear program.

The LP recourse problem applies to two-stage optimization problems where uncertainty in resource availability of the second stage hinders informed decision making. The recourse function affords a way to compensate "later" for an error in prediction "now." The literature provides a rich body of work on the optimization of such problems, but little research has been accomplished regarding the characterization of the surface in the local region of optimality, in particular sensitivity analysis. A decision maker faced with considerations other than the modeled objective function must be presented with a way to estimate the impact of operating at non-optimal decision variable values. This work develops and demonstrates a technique for characterizing the surface using response surface methodology. Specifically, the flexibility and utility of RSM techniques applied to this class of problems is demonstrated, and a methodology for characterizing the surface in the local region using a low-order polynomial is developed.

This article considers the problem of bounding the expected value of a linear program (LP) containing random coefficients, with applications to solving two-stage stochastic programs. An upper bound for minimizations is derived from a restriction of an equivalent, penalty-based formulation of the primal stochastic LP, and a lower bound is obtained from a restriction of a reformulation of the dual. These restricted-recourse bounds are more general and more easily computed than most other bounds because random coefficients may appear anywhere in the LP, neither independence nor boundedness of the coefficients is needed, and the bound is computed by solving a single LP or nonlinear program. Analytical examples demonstrate that the new bounds can be stronger than complementary Jensen bounds. (An upper bound is complementary to a lower bound, and vice versa). In computational work, the authors apply the bounds to a two-stage stochastic program for semiconductor manufacturing with uncertain demand and production rates.

This research investigated an alternative to the traditional approaches of optimizing a stochastic response surface subject to constraints. This research investigated the bias in the expected value of the solution, possible alternative decision variable settings, and a method to improve the solution. A three step process is presented to evaluate stochastic response surfaces subject to constraints. Step 1 is to use a traditional approach to estimate the response surface and a covariance matrix through regression. Step 2 samples the objective function of the linear program (i.e., response surface) and identifies the extreme points visited. Step 3 presents a method to estimate the optimal extreme point and present that information to a decision maker.

Results and problems in the ordinary 'here-and-now' and 'wait-and- see' stochastic linear programming problems are described. A general formulation of the 'here-and-now' problem is presented, and an approach for solving a special kind of 'here-and-now' problem is suggested.

Traditional Linear Genetic Programming (LGP) algorithms are based only on the selection mechanism to guide the search. Genetic operators combine or mutate random portions of the individuals, without knowing if the result will lead to a fitter individual. Probabilistic Model Building Genetic Programming (PMB-GP) methods were proposed to overcome this issue through a probability model that captures the structure of the fit individuals and use it to sample new individuals. This work proposes the use of LGP with a Stochastic Context-Free Grammar (SCFG), that has a probability distribution that is updated according to selected individuals. We proposed a method for adapting the grammar into the linear representation of LGP. Tests performed with the proposed probabilistic method, and with two hybrid approaches, on several symbolic regression benchmark problems show that the results are statistically better than the obtained by the traditional LGP.

Traditional Linear Genetic Programming (LGP) algorithms are based only on the selection mechanism to guide the search. Genetic operators combine or mutate random portions of the individuals, without knowing if the result will lead to a fitter individual. Probabilistic Model Building Genetic Programming (PMB-GP) methods were proposed to overcome this issue through a probability model that captures the structure of the fit individuals and use it to sample new individuals. This work proposes the use of LGP with a Stochastic Context-Free Grammar (SCFG), that has a probability distribution that is updated according to selected individuals. We proposed a method for adapting the grammar into the linear representation of LGP. Tests performed with the proposed probabilistic method, and with two hybrid approaches, on several symbolic regression benchmark problems show that the results are statistically better than the obtained by the traditional LGP.

We are concerned with the linear-quadratic optimal stochastic control problem with random coefficients. Under suitable conditions, we prove that the value field $V(t,x,\omega), (t,x,\omega)\in [0,T]\times R^n\times \Omega$, is quadratic in $x$, and has the following form: $V(t,x)=\langle K_tx, x\rangle$ where $K$ is an essentially bounded nonnegative symmetric matrix-valued adapted processes. Using the dynamic programming principle (DPP), we prove that $K$ is a continuous semi-martingale of the form $$K_t=K_0+\int_0^t \, dk_s+\sum_{i=1}^d\int_0^tL_s^i\, dW_s^i, \quad t\in [0,T]$$ with $k$ being a continuous process of bounded variation and $$E\left[\left(\int_0^T|L_s|^2\, ds\right)^p\right]

We consider the infinite dimensional linear programming (inf-LP) approach for solving stochastic control problems. The inf-LP corresponding to problems with uncountable state and input spaces is in general computationally intractable. By focusing on linear systems with quadratic cost (LQG), we establish a connection between this approach and the well-known Riccati LMIs. In particular, we show that the semidefinite programs known for the LQG problem can be derived from the pair of primal and dual inf-LPs. Furthermore, we establish a connection between multi-objective and chance constraint criteria and the inf-LP formulation.

The paper gives an algorithm for L-shaped linear programs which arise naturally in optimal control problems with state constraints and stochastic linear programs (which can be represented in this form with an infinite number of linear constraints). The first section describes a cutting hyperplane algorithm which is shown to be equivalent to a partial decomposition algorithm of the dual program. The two last sections are devoted to applications of the cutting hyperplane algorithm to a linear optimal control problem and stochastic programming problems.

This paper presents a simple method of allowing for uncertainties in the constant terms (i.e. right hand sides) of a linear programming problem, and hence producing realistic 'safety margins' in the solution. This is done by fitting a mixture of uniform distributions to the assumed distributions of these right hand sides, and using a particular quadratic programming algorithm.

In this paper, we consider deterministic models for a stochastic linear program with a constant feasible region and stochastic cost coefficients having multi-variate normal distribution. Relationships among the solutions of these models are examined and it is shown that solving a parametric quadratic program associated with Markowitz's mean-variance model yields solutions to all other models considered for all relevant values of parameters. (Author)

In this paper, we consider deterministic models for a stochastic linear program with a constant feasible region and stochastic cost coefficients having multi-variate normal distribution. Relationships among the solutions of these models are examined and it is shown that solving a parametric quadratic program associated with Markowitz's mean-variance model yields solutions to all other models considered for all relevant values of parameters. (Author)

In this paper, we consider deterministic models for a stochastic linear program with a constant feasible region and stochastic cost coefficients having multi-variate normal distribution. Relationships among the solutions of these models are examined and it is shown that solving a parametric quadratic program associated with Markowitz's mean-variance model yields solutions to all other models considered for all relevant values of parameters. (Author)

Use of Chance-Constrained Programming to Account for Stochastic Variation in the A-Matrix of Large-Scale Linear Programs: A Forestry Application

Multi-stage stochastic linear programs (MSLPs) are notoriously hard to solve in general. Linear decision rules (LDRs) yield an approximation of an MSLP by restricting the decisions at each stage to be an affine function of the observed uncertain parameters. Finding an optimal LDR is a static optimization problem that provides an upper bound on the optimal value of the MSLP, and, under certain assumptions, can be formulated as an explicit linear program. Similarly, as proposed by Kuhn, Wiesemann, and Georghiou (Math. Program., 130, 177-209, 2011) a lower bound for an MSLP can be obtained by restricting decisions in the dual of the MSLP to follow an LDR. We propose a new approximation approach for MSLPs, two-stage LDRs. The idea is to require only the state variables in an MSLP to follow an LDR, which is sufficient to obtain an approximation of an MSLP that is a two-stage stochastic linear program (2SLP). We similarly propose to apply LDR only to a subset of the variables in the dual of the MSLP, which yields a 2SLP approximation of the dual that provides a lower bound on the optimal value of the MSLP. Although solving the corresponding 2SLP approximations exactly is intractable in general, we investigate how approximate solution approaches that have been developed for solving 2SLP can be applied to solve these approximation problems, and derive statistical upper and lower bounds on the optimal value of the MSLP. In addition to potentially yielding better policies and bounds, this approach requires many fewer assumptions than are required to obtain an explicit reformulation when using the standard static LDR approach. As an illustrative example we apply our approach to a capacity expansion model, and find that the two-stage LDR policy has expected cost between 20% and 34% lower than the static LDR policy, and in the dual yields lower bounds that are between 0.1% and 3.3% better.