Solving non-linear portfolio optimization problems with the primal-dual interior point method

Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear or quadratic models due to the inability of currently available solvers to solve NLP problems of typical sizes. However stochastic programming problems are highly structured. The key to the efficient solution of such problems is therefore the ability to exploit their structure. Interior point methods are well-suited to the solution of very large non-linear optimization problems. In this paper we exploit this feature and show how portfolio optimization problems with sizes measured in millions of constraints and decision variables, featuring constraints on semi-variance, skewness or non-linear utility functions in the objective, can be solved with the state-of-the-art solver.     

<Emphasis FontCategory="NonProportional">pyomo.dae</Emphasis>: a modeling and automatic discretization framework for optimization with differential and algebraic equations

We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differential equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.     

PySP: modeling and solving stochastic programs in Python

Although stochastic programming is a powerful tool for modeling decision-making under uncertainty, various impediments have historically prevented its wide-spread use. One factor involves the ability of non-specialists to easily express stochastic programming problems as extensions of their deterministic counterparts, which are typically formulated first. A second factor relates to the difficulty of solving stochastic programming models, particularly in the mixed-integer, non-linear, and/or multi-stage cases. Intricate, configurable, and parallel decomposition strategies are frequently required to achieve tractable run-times on large-scale problems. We simultaneously address both of these factors in our PySP software package, which is part of the Coopr open-source Python repository for optimization; the latter is distributed as part of IBM’s COIN-OR repository. To formulate a stochastic program in PySP, the user specifies both the deterministic base model (supporting linear, non-linear, and mixed-integer components) and the scenario tree model (defining the problem stages and the nature of uncertain parameters) in the Pyomo open-source algebraic modeling language. Given these two models, PySP provides two paths for solution of the corresponding stochastic program. The first alternative involves passing an extensive form to a standard deterministic solver. For more complex stochastic programs, we provide an implementation of Rockafellar and Wets’ Progressive Hedging algorithm. Our particular focus is on the use of Progressive Hedging as an effective heuristic for obtaining approximate solutions to multi-stage stochastic programs. By leveraging the combination of a high-level programming language (Python) and the embedding of the base deterministic model in that language (Pyomo), we are able to provide completely generic and highly configurable solver implementations. PySP has been used by a number of research groups, including our own, to rapidly prototype and solve difficult stochastic programming problems.     

Applying the progressive hedging algorithm to stochastic generalized networks

The introduction of uncertainty to mathematical programs greatly increases the size of the resulting optimization problems. Specialized methods that exploit program structures and advances in computer technology promise to overcome the computational complexity of certain classes of stochastic programs. In this paper we examine the progressive hedging algorithm for solving multi-scenario generalized networks. We present computational results demonstrating the effect of various internal tactics on the algorithm's performance. Comparisons with alternative solution methods are provided.Research supported in part by grants from the National Science Foundation (DCR-861-4057) and the Mathematical and Analytics Computation Center of IBM Corporation, New York.     

Accelerating Branch-and-Bound through a Modeling Language Construct for Relaxation-Specific Constraints

In the tradition of modeling languages for optimization, a single model is passed to a solver for solution. In this paper, we extend BARON’s modeling language in order to facilitate the communication of problem-specific relaxation information from the modeler to the branch-and-bound solver. This effectively results into two models being passed from the modeling language to the solver. Three important application areas are identified and computational experiments are presented. In all cases, nonlinear constraints are provided only to the relaxation constructor in order to strengthen the lower bounding step of the algorithm without complicating the local search process. In the first application area, nonlinear constraints from the reformulation–linearization technique (RLT) are added to strengthen a problem formulation. This approach is illustrated for the pooling problem and computational results show that it results in a scheme that makes global optimization nearly as fast as local optimization for pooling problems from the literature. In the second application area, we communicate with the relaxation constructor the first-order optimality conditions for unconstrained global optimization problems. Computational experiments with polynomial programs demonstrate that this approach leads to a significant reduction of the size of the branch-and-bound search tree. In the third application, problem-specific nonlinear optimality conditions for the satisfiability problem are used to strengthen the lower bounding step and are found to significantly expedite the branch-and-bound algorithm when applied to a nonlinear formulation of this problem.     

A comparison of mathematical programming modeling systems

We compare three mathematical programming modeling languages, GAMS, OMNI and MathPro. To understand the properties of these languages, we formulate four linear programs in each language. The formulations are representative of the kinds of model structures one encounters in practice. Each of the languages focuses on a different view of linear programs. GAMS approximates algebra, OMNI uses the activity view and MathPro uses a block schematic. We summarize our experiences with the languages and suggest areas for further enhancement.     

An efficient grouping genetic algorithm for U-shaped assembly line balancing problems with maximizing production rate

U-type assembly line is one of the important tools that may increase companies’ production efficiency. In this study, two different modeling approaches proposed for the assembly line balancing problems have been used in modeling type-II U-line balancing problems, and the performances of these models have been compared with each other. It has been shown that using mathematical formulations to solve medium and large size problem instances is impractical since the problem is NP-hard. Therefore, a grouping genetic and simulated annealing algorithms have been developed, and a particle swarm optimization algorithm is adapted to compare with the proposed methods. A special crossover operator that always obtains feasible offspring has been suggested for the proposed grouping genetic algorithm. Furthermore, a local search procedure based on problem-specific knowledge was applied to increase the intensification of the algorithm. A set of well-known benchmark instances was solved to evaluate the effectiveness of the proposed and existing methods. Results showed that while the mathematical formulations can only be used to solve small size instances, metaheuristics can obtain high quality solutions for all size problem instances within acceptable CPU times. Moreover, grouping genetic algorithm has been found to be superior to the other methods according to the number of optimal solutions, or deviations from the lower bound values.     

Opticon: An algorithm for the optimal control of nonlinear stochastic models

In this paper we describe the algorithm OPTCON which has been developed for the optimal control of nonlinear stochastic models. It can be applied to obtain approximate numerical solutions of control problems where the objective function is quadratic and the dynamic system is nonlinear. In addition to the usual additive uncertainty, some or all of the parameters of the model may be stochastic variables. The optimal values of the control variables are computed in an iterative fashion: First, the time-invariant nonlinear system is linearized around a reference path and approximated by a time-varying linear system. Second, this new problem is solved by applying Bellman's principle of optimality. The resulting feedback equations are used to project expected optimal state and control variables. These projections then serve as a new reference path, and the two steps are repeated until convergence is reached. The algorithm has been implemented in the statistical programming system GAUSS. We derive some mathematical results needed for the algorithm and give an overview of the structure of OPTCON. Moreover, we report on some tentative applications of OPTCON to two small macroeconometric models for Austria.     

New constructs for the description of combinatorial optimization problems in algebraic modeling languages

Algebraic languages are at the heart of many successful optimization modeling systems, yet they have been used with only limited success for combinatorial (or discrete) optimization. We show in this paper, through a series of examples, how an algebraic modeling language might be extended to help with a greater variety of combinatorial optimization problems. We consider specifically those problems that are readily expressed as the choice of a subset from a certain set of objects, rather than as the assignment of numerical values to variables. Since there is no practicable universal algorithm for problems of this kind, we explore a hybrid approach that employs a general-purpose subset enumeration scheme together with problem-specific directives to guide an efficient search.     

Integer linear programming models for grid-based light post location problem

Selecting optimal location is a key decision problem in business and engineering. This research focuses to develop mathematical models for a special type of location problems called grid-based location problems. It uses a real-world problem of placing lights in a park to minimize the amount of darkness and excess supply. The non-linear nature of the supply function (arising from the light physics) and heterogeneous demand distribution make this decision problem truly intractable to solve. We develop ILP models that are designed to provide the optimal solution for the light post problem: the total number of light posts, the location of each light post, and their capacities (i.e., brightness). Finally, the ILP models are implemented within a standard modeling language and solved with the CPLEX solver. Results show that the ILP models are quite efficient in solving moderately sized problems with a very small optimality gap.     

OSiL: An instance language for optimization

Distributed computing technologies such as Web Services are growing rapidly in importance in today’s computing environment. In the area of mathematical optimization, it is common to separate modeling languages from optimization solvers. In a completely distributed environment, the modeling language software, solver software, and data used to generate a model instance might reside on different machines using different operating systems. Such a distributed environment makes it critical to have an open standard for exchanging model instances. In this paper we present OSiL (Optimization Services instance Language), an XML-based computer language for representing instances of large-scale optimization problems including linear programs, mixed-integer programs, quadratic programs, and very general nonlinear programs. OSiL has two key features that make it much superior to current standard forms for optimization problem instances. First, it uses the object-oriented features of XML schemas to efficiently represent nonlinear expressions. Second, its XML schema maps directly into a corresponding in-memory representation of a problem instance. The in-memory representation provides a robust application program interface for general nonlinear programming, facilitates reading and writing postfix, prefix, and infix formats to and from the nonlinear expression tree, and makes the expression tree readily available for function and derivative evaluations.     

Embedding optimisation algorithms with Mosel

Algebraic modelling languages allow models to be implemented in such a way that they can easily be understood and modified. They are therefore a working environment commonly used by practitioners in Operations Research. Having once developed models, they need to be integrated inside the company information system. This step often involves embedding a model into a programming language environment: many existing algebraic modelling languages make possible to run parameterised models and subsequently retrieve their results, but without any facility for interacting with the model during the model generation or solution process.In this paper we show how to use the Mosel environment to implement complex algorithms directly in the modelling language.The Office cleaning problem is solved by a branch-and-cut algorithm, implemented entirely in the modelling language (including the definition of the callback function for the solver). Secondly, a cutting stock problem is solved by column generation, also implemented in the modelling language.AMS classification: 90Cxx, 65K05, 68N15     

Some recent progress in applied mathematics (Ⅰ)

Some activities of applied mathematics are seen from different angles in a two part series. In part 1, we will emphasize on mathematical modeling, exponential prediction models. On a systemic construction of the quantitative mathematics, it is shown that there is an impassible chasm between pure and applied mathematics, that existance and structures of systems are independent of human consciousness. For the purpose of prediction, concepts of calculus are generalized to discrete time series.     

应用数学方面一些最新的进展

Some activities of applied mathematics are seen from different angles in a two-part series. In part 1, we will emphasize on mathematical modeling, exponential prediction models. On a systemic construction of the quantitative mathematics,it is shown that there is an impassible chasm between pure and applied mathematics, that existance and structures of systems are independent of human consciousness. For the purpose of prediction, concepts of calculus are generalized to discrete time series.     

Mathematical Programming Modelling Using MGG

MGG is a software package for the application of mathematicalprogramming (MP). It complements the Sciconic MP code by providinga facility for developing MP models quickly and efficiently,and it enables changes to be made easily to established models.It is available on a wide range of minis and mainframes anda version of it is available as part of the Micro LP systemon IBM PCs and compatibles. MGG is based on an approach to modellingMP problems which stresses the primacy of the mathematical formulation.The process of MP modelling is divided into two stages: modelpreparation and running of the model. The user writes a mathematicalformulation, which MGG converts to matrix generator and reportwriter programs. This is done once to produce the programs whichcan then be run many times on different data. This paper describesMGG and draws comparisons with other matrix generator and mathematicalprogramming languages. It starts by considering how an MP problemcan be described, and then sets out a methodology for formulation.The MGG language is based upon this approach. A simple exampleis presented which is shown both as an algebraic formulationand in the MGG language. The process of building and runningmodels with MGG is then described. Finally, some comments areoffered on experience of using the software.     

Decision-dependent probabilities in stochastic programs with recourse

Stochastic programming with recourse usually assumes uncertainty to be exogenous. Our work presents modelling and application of decision-dependent uncertainty in mathematical programming including a taxonomy of stochastic programming recourse models with decision-dependent uncertainty. The work includes several ways of incorporating direct or indirect manipulation of underlying probability distributions through decision variables in two-stage stochastic programming problems. Two-stage models are formulated where prior probabilities are distorted through an affine transformation or combined using a convex combination of several probability distributions. Additionally, we present models where the parameters of the probability distribution are first-stage decision variables. The probability distributions are either incorporated in the model using the exact expression or by using a rational approximation. Test instances for each formulation are solved with a commercial solver, BARON, using selective branching.     

Logic-Based Modeling and Solution of Nonlinear Discrete/Continuous Optimization Problems

This paper presents a review of advances in the mathematical programming approach to discrete/continuous optimization problems. We first present a brief review of MILP and MINLP for the case when these problems are modeled with algebraic equations and inequalities. Since algebraic representations have some limitations such as difficulty of formulation and numerical singularities for the nonlinear case, we consider logic-based modeling as an alternative approach, particularly Generalized Disjunctive Programming (GDP), which the authors have extensively investigated over the last few years. Solution strategies for GDP models are reviewed, including the continuous relaxation of the disjunctive constraints. Also, we briefly review a hybrid model that integrates disjunctive programming and mixed-integer programming. Finally, the global optimization of nonconvex GDP problems is discussed through a two-level branch and bound procedure.     

Two stage linear programming under uncertainty with 0–1 integer first stage variables

Stochastic programs with continuous variables are often solved using a cutting plane method similar to Benders' partitioning algorithm. However, mixed 0–1 integer programs are also solved using a similar procedure along with enumeration. This similarity is exploited in this paper to solve two stage linear programs under uncertainty where the first stage variables are 0–1. Such problems often arise in capital investment. A network investment application is given which includes as a special case a coal transportation problem.