Probabilistic inequality constraints in stochastic optimal control theory

There are many optimal control problems in which it is necessary or desirable to constrain the range of values of state variables. When stochastic inputs are involved, these inequality constraint problems are particularly difficult. Sometimes the constraints must be modeled as hard constraints which can never be violated, and other times it is more natural to prescribe a probability that the constraints will not be violated. This paper treats general problems of the latter type, in which probabilistic inequality constraints are imposed on the state variables or on combinations of state and control variables. A related class of problems in which the state is required to reach a target set with a prescribed probability is handled by the same methods. It is shown that the solutions to these problems can be obtained by solving a comparatively simple bilinear deterministic control problem.     

Generalized functions which are analytic with respect to some of their variables

For a generalized function which is analytic with respect to some of its variables in the sense of Gording and Malgrange we prove a theorem on representation in the form of a power series in these variables which converges in D′ and a corresponding uniqueness theorem. We also obtain necessary and sufficient conditions for a generalized function to be entire with respect to some of its variables.     

Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems

Numerous planning problems can be formulated as multi-stage stochastic programs and many possess key discrete (integer) decision variables in one or more of the stages. Progressive hedging (PH) is a scenario-based decomposition technique that can be leveraged to solve such problems. Originally devised for problems possessing only continuous variables, PH has been successfully applied as a heuristic to solve multi-stage stochastic programs with integer variables. However, a variety of critical issues arise in practice when implementing PH for the discrete case, especially in the context of very difficult or large-scale mixed-integer problems. Failure to address these issues properly results in either non-convergence of the heuristic or unacceptably long run-times. We investigate these issues and describe algorithmic innovations in the context of a broad class of scenario-based resource allocation problem in which decision variables represent resources available at a cost and constraints enforce the need for sufficient combinations of resources. The necessity and efficacy of our techniques is empirically assessed on a two-stage stochastic network flow problem with integer variables in both stages.     

Worst-case global optimization of black-box functions through Kriging and relaxation

A new algorithm is proposed to deal with the worst-case optimization of black-box functions evaluated through costly computer simulations. The input variables of these computer experiments are assumed to be of two types. Control variables must be tuned while environmental variables have an undesirable effect, to which the design of the control variables should be robust. The algorithm to be proposed searches for a minimax solution, i.e., values of the control variables that minimize the maximum of the objective function with respect to the environmental variables. The problem is particularly difficult when the control and environmental variables live in continuous spaces. Combining a relaxation procedure with Kriging-based optimization makes it possible to deal with the continuity of the variables and the fact that no analytical expression of the objective function is available in most real-case problems. Numerical experiments are conducted to assess the accuracy and efficiency of the algorithm, both on analytical test functions with known results and on an engineering application.     

On the transitivity of the comonotonic and countermonotonic comparison of random variables

A recently proposed method for the pairwise comparison of arbitrary independent random variables results in a probabilistic relation. When restricted to discrete random variables uniformly distributed on finite multisets of numbers, this probabilistic relation expresses the winning probabilities between pairs of hypothetical dice that carry these numbers and exhibits a particular type of transitivity called dice-transitivity. In case these multisets have equal cardinality, two alternative methods for statistically comparing the ordered lists of the numbers on the faces of the dice have been studied recently: the comonotonic method based upon the comparison of the numbers of the same rank when the lists are in increasing order, and the countermonotonic method, also based upon the comparison of only numbers of the same rank but with the lists in opposite order. In terms of the discrete random variables associated to these lists, these methods each turn out to be related to a particular copula that joins the marginal cumulative distribution functions into a bivariate cumulative distribution function. The transitivity of the generated probabilistic relation has been completely characterized. In this paper, the list comparison methods are generalized for the purpose of comparing arbitrary random variables. The transitivity properties derived in the case of discrete uniform random variables are shown to be generic. Additionally, it is shown that for a collection of normal random variables, both comparison methods lead to a probabilistic relation that is at least moderately stochastic transitive.     

Newsvendor-type models with decision-dependent uncertainty

Models for decision-making under uncertainty use probability distributions to represent variables whose values are unknown when the decisions are to be made. Often the distributions are estimated with observed data. Sometimes these variables depend on the decisions but the dependence is ignored in the decision maker??s model, that is, the decision maker models these variables as having an exogenous probability distribution independent of the decisions, whereas the probability distribution of the variables actually depend on the decisions. It has been shown in the context of revenue management problems that such modeling error can lead to systematic deterioration of decisions as the decision maker attempts to refine the estimates with observed data. Many questions remain to be addressed. Motivated by the revenue management, newsvendor, and a number of other problems, we consider a setting in which the optimal decision for the decision maker??s model is given by a particular quantile of the estimated distribution, and the empirical distribution is used as estimator. We give conditions under which the estimation and control process converges, and show that although in the limit the decision maker??s model appears to be consistent with the observed data, the modeling error can cause the limit decisions to be arbitrarily bad.     

Explicit Formulas for Generalized Action–Angle Variables in a Neighborhood of an Isotropic Torus and Their Application

Different versions of the Darboux–Weinstein theorem guarantee the existence of action–angle-type variables and the harmonic-oscillator variables in a neighborhood of isotropic tori in the phase space. The procedure for constructing these variables is reduced to solving a rather complicated system of partial differential equations. We show that this system can be integrated in quadratures, which permits reducing the problem of constructing these variables to solving a system of quadratic equations. We discuss several applications of this purely geometric fact in problems of classical and quantum mechanics.     

Quaternionic hyperfunctions on five-dimensional varieties in ℍ2

In this article we introduce a new notion of differential forms to describe the cohomology associated to the sheaf of regular functions in several quaternionic variables. We then use these differential forms to introduce and describe concretely a sheaf of quaternionic hyperfunctions as boundary values of regular functions in two quaternionic variables. We show how these ideas can be generalized to the case of monogenic functions in two vector variables with values in a Clifford algebra.     

On maintenance scheduling of production units

A typical maintenance scheduling problem is presented as a large-scale mixed integer nonlinear programming case. Several relaxations of the conditions of variables and constraints are discussed. The optimal solution of the models based on these relaxations is viewed as the lower bound of the optimal solution in the original problem. A combined implicit enumeration and branch-and-bound algorithm is used. Typical dimension of the problems for which computational experience is reported is 25 production units in the system. 19 of these are to be maintained and a planning horizon of 52 weeks with 5 types of hours per week. The corresponding dimensions of the model are about 5700 constraints, 700 binary variables and 6500 nonlinear separable variables.     

Interpreting latent variables in factor models via convex optimization

Latent or unobserved phenomena pose a significant difficulty in data analysis as they induce complicated and confounding dependencies among a collection of observed variables. Factor analysis is a prominent multivariate statistical modeling approach that addresses this challenge by identifying the effects of (a small number of) latent variables on a set of observed variables. However, the latent variables in a factor model are purely mathematical objects that are derived from the observed phenomena, and they do not have any interpretation associated to them. A natural approach for attributing semantic information to the latent variables in a factor model is to obtain measurements of some additional plausibly useful covariates that may be related to the original set of observed variables, and to associate these auxiliary covariates to the latent variables. In this paper, we describe a systematic approach for identifying such associations. Our method is based on solving computationally tractable convex optimization problems, and it can be viewed as a generalization of the minimum-trace factor analysis procedure for fitting factor models via convex optimization. We analyze the theoretical consistency of our approach in a high-dimensional setting as well as its utility in practice via experimental demonstrations with real data.     

An imbedding theorem in the calculus of variations for multiple integrals,addendum

It is shown how a set of canonical variables in the sense of Rund [5] can be associated with a given extremal of a multiple integral variational problem in a simple, direct manner. The definition of these variables in a previous paper [1], which is concerned with the problem of imbedding a given extremal in anr-geodesic field, is thereby clarified and abbreviated considerably. A theorem due essentially to Boerner, which is crucial to the imbedding theorem given in [1], is proved more easily and under less restrictive hypotheses than in [1]. Furthermore, it is shown how the present definition of the canonical variables allows one to eliminate from the geodesic field theory of Carathéodory the restriction that the Lagrangian be non-vanishing along the extremal.     

A compact formulation of an elastoplastic analysis problem

Two important problems in the area of engineering plasticity are limit load analysis and elastoplastic analysis. It is well known that these two problems can be formulated as linear and quadratic programming problems, respectively (Refs. 1–2). In applications, the number of variables in each of these mathematical programming problems tends to be large. Consequently, it is important to have efficient numerical methods for their solution. The purpose of this paper is to present a method which allows the quadratic programming formulation of the elastoplastic analysis to be reformulated as an equivalent quadratic programming problem which has significantly fewer variables than the original formulation. Indeed, in Section 4, we will present details of an example for which the original quadratic programming formulation required 297 variables and for which the equivalent formulation presented here required only two variables. The method is based on a characterization of the entire family of optimal solutions for a linear programming problem.This research was supported by the Natural Science and Engineering Council of Canada under Grant No. A8189 and by a Leave Fellowship from the Social Sciences and Humanities Research Council of Canada. The author takes pleasure in acknowledging many stimulating discussions with Professor D. E. Grierson.     

Modeling disjunctive constraints with a logarithmic number of binary variables and constraints

Many combinatorial constraints over continuous variables such as SOS1 and SOS2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of a finite number of specially structured polyhedra. Known mixed integer binary formulations for these constraints have a number of binary variables and extra constraints linear in the number of polyhedra. We give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints logarithmic in the number of polyhedra. Using these conditions we introduce mixed integer binary formulations for SOS1 and SOS2 constraints that have a number of binary variables and extra constraints logarithmic in the number of continuous variables. We also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints logarithmic in the number of linear pieces of the functions. We prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations.     

Clusterwise analysis for multiblock component methods

Multiblock component methods are applied to data sets for which several blocks of variables are measured on a same set of observations with the goal to analyze the relationships between these blocks of variables. In this article, we focus on multiblock component methods that integrate the information found in several blocks of explanatory variables in order to describe and explain one set of dependent variables. In the following, multiblock PLS and multiblock redundancy analysis are chosen, as particular cases of multiblock component methods when one set of variables is explained by a set of predictor variables that is organized into blocks. Because these multiblock techniques assume that the observations come from a homogeneous population they will provide suboptimal results when the observations actually come from different populations. A strategy to palliate this problem—presented in this article—is to use a technique such as clusterwise regression in order to identify homogeneous clusters of observations. This approach creates two new methods that provide clusters that have their own sets of regression coefficients. This combination of clustering and regression improves the overall quality of the prediction and facilitates the interpretation. In addition, the minimization of a well-defined criterion—by means of a sequential algorithm—ensures that the algorithm converges monotonously. Finally, the proposed method is distribution-free and can be used when the explanatory variables outnumber the observations within clusters. The proposed clusterwise multiblock methods are illustrated with of a simulation study and a (simulated) example from marketing.     

Unrestricted variables in linear programming

There are a number of ways of dealing with a linear programming problem in which some variables are allowed to take on negative values. One of these methods, which is either ignored or mentioned only incidentally in most textbooks, requires only one additional variable to be introduced regardless of how many unrestricted variables the original problem has. In this note, this method is interpreted geometrically and an application to the computation of extreme points is given.     

Autonomous Planar Systems of Riccati Type

The role of Riccati type systems in the plane along with the related linear, second order differential equation is examined. If $x$ and $y$ are the variables of the Riccati differential equation, then any integrable Riccati system has two independent invariant curves dependent upon these variables whose nature is easily determined from the solution of the linear equation. Each of these curves has the same cofactor. Other invariant curves depend upon $x$ alone and are shown to be less important. The systems have both Liouvillian and non--Liouvillian solutions and are easily transformable to symmetric systems. However, systems derived from them may not be symmetric in their transformed variables. Several systems from the literature are discussed with regard to the forms of the invariant curves presented in the paper. The relation of certain Riccati type systems is considered with respect to Abel differential equations.     

Stability of Richtmyer Type Difference Schemes in any Finite Number of Space Variables and Their Comparison with Multistep Strang Schemes

The stability of generalized Richtmyer two step difference schemesin any finite number of space variables is examined and a sufficientstability condition obtained for each scheme. In certain casesthis condition is shown to be optimal C.F.L. The efficiencyof these schemes in solving time dependent problems in two andthree space variables is examined and the schemes are seen tocompare favourably with the corresponding multistep forms ofStrang's schemes.     

Retail credit stress testing using a discrete hazard model with macroeconomic factors

Retail credit models are implemented using discrete survival analysis, enabling macroeconomic conditions to be included as time-varying covariates. In consequence, these models can be used to estimate changes in probability of default given downturn economic scenarios. Compared with traditional models, we offer improved methodologies for scenario generation and for the use of them to predict default rates. Monte Carlo simulation is used to generate a distribution of estimated default rates from which Value at Risk and Expected Shortfall are computed as a means of stress testing. Several macroeconomic variables are considered and in particular factor analysis is employed to model the structure between these variables. Two large UK data sets are used to test this approach, resulting in plausible dynamic models and stress test outcomes.     

Adding activities to the dual instead of cuts to the primal problem

When solving a problem by appending cuts the dimension of the corresponding simplex tableau and the basic inverse oscillates, which makes it difficult to implement a cutting plane algorithm based on a standard LP code. Moreover, it is complicated to express a cut in the original variables. In this paper we show that by formulating the dual to the problem and adding activities, these adverse effects can be circumvented. It is shown that the set of activities which can be added is the same as the set of cuts which can be appended and that it is easy to exhibit an activity in the original primal variables. As a consequence of this a new formulation of a cut in the original primal variables is given.     

Global optimization of signomial mixed-integer nonlinear programming problems with free variables

Mixed-integer nonlinear programming (MINLP) problems involving general constraints and objective functions with continuous and integer variables occur frequently in engineering design, chemical process industry and management. Although many optimization approaches have been developed for MINLP problems, these methods can only handle signomial terms with positive variables or find a local solution. Therefore, this study proposes a novel method for solving a signomial MINLP problem with free variables to obtain a global optimal solution. The signomial MINLP problem is first transformed into another one containing only positive variables. Then the transformed problem is reformulated as a convex mixed-integer program by the convexification strategies and piecewise linearization techniques. A global optimum of the signomial MINLP problem can finally be found within the tolerable error. Numerical examples are also presented to demonstrate the effectiveness of the proposed method.