Darinka Dentcheva Stevens Institute of Technology, Hoboken NJ, USA »homepage »Asymptotics of Optimization Problems with Composite Risk Functionals
Laureano F. Escudero Universidad Rey Juan Carlos, Móstoles (Madrid), Spain »homepage »New trends on Matheuristic Algorithms for Mathematical Optimization under Uncertainty. Some results
Jiří Jaromír Klemeš Brno University of Technology, Brno, Czech Republic Pázmány Péter Catholic University, Budapest, Hungary »homepage »Mathematical Programming and Graph-Based Tools in Process Systems Engineering
Ulrike Leopold-Wildburger University of Graz, Graz, Austria »homepage »Optimization and behavior
Andrzej Ruszczynski Rutgers University, Piscataway NJ, USA »homepage »Risk-Averse Control of Markov System
Ana Viana Centre for Industrial Engineering and Management of INESC TEC, Porto, Portugal »homepage »Optimization challenges in Kidney Exchange Programs: past, present and future

Asymptotics of Optimization Problems with Composite Risk Functionals

Darinka Dentcheva, Stevens Institute of Technology, Hoboken NJ, USA

Abstract

Risk quantification and risk management in finance, insurance and other areas have attracted a lot of attention among scientists and practitioners due to the practical relevance and theoretical challenges they present. Mathematical models of risk lead to new structures in convex analysis, optimization, optimal control, and statistics. Risk models evaluate gains or losses depending on a decision maker's choice z and random quantities, which may be summarized in a random vector X. We are interested in a functional f(z,X), which may be optimized under practically relevant restrictions on the decisions z. Next to some moments of the random variable Y= f(z,X), very frequently, models of risk use a nonlinear functionals of the distribution of Y. In practice, we can only use observations to estimate the model. Several measures of risk have an explicit formula, which can be used as a plug-in estimator, with the original measure P replaced by the empirical measure. The questions pertaining to statistical estimation of risk functionals and its effect on optimization problems using such functionals are crucial for practical application. The composite structure and limited differentiability properties of coherent measures of risk render the existing theory inapplicable.

In this talk, we discuss statistical estimation of composite risk mappings depending on random vectors and their moments. We present a central limit formula for such functionals and provide a characterization of the limiting distribution of the empirical estimators. Several popular risk measures will be presented as illustrative examples. While we show that many known coherent measures of risk can be cast in the presented structures, we emphasize that the results are of more general nature with a potentially wider applicability.

Additionally, we consider sample based optimization problems in which composite risk functionals are used as objectives or constraints. We characterize the asymptotic behavior of the optimal value and the optimal solutions of the problems.

Applications of the results to hypothesis testing of stochastic orders and portfolio efficiency will be outlined.

New trends on Matheuristic Algorithms for Mathematical Optimization under Uncertainty. Some results

Laureano F. Escudero, Universidad  Rey Juan Carlos, Móstoles (Madrid), Spain 

Abstract

Given the huge dimensions (very frequently, up to hundreds of thousands of constraints and variables) of multiperiod stochastic mixed 0-1 models to deal with in practical applications, it seems unrealistic to seek for optimal solutions, although the scheme for   guaranteeing the solution’s quality is a must.  A review of new matheuristic algorithms in the literature is presented for the risk neutral model as well as for considering some risk averse measures, especially stochastic dominance –based ones. A good tendency in this type of research lies on matheuristic versions of the Nested Stochastic Decomposition (NSD) methodology for while considering period wise dependent uncertainty for solving large-scale dynamic mixed 0-1 problems. One of the main reasons for its good performance is that the partition of the periods in stages (of consecutive stages) makes the NSD decomposition procedure easier, where each iteration has forward and backward steps. Notice that the constraint system of any stage submodel includes independent submodels in its risk neutral version.  Each submodel is supported by a subtree that is rooted with a strategic node includes also the set of immediate successor nodes of each leaf strategic node in the previous stage. So, those submodels can be solved in parallel, whose their good performance requires some type of communication between them.  However, in spite of the advantages of using the NSD methodology for dynamic problem solving, it has still some drawbacks. In fact, it can be observed in computational experience with stochastic mixed 0-1 models that the NSD's efficiency is reduced for those problems with stepwise dependent non-Markovian processes where the state variables link several consecutive stages. Some modeling hints are given for reducing it down to two consecutive stages, if possible. Versions of the NSD for expected conditional stochastic dominance functional will also be presented. Computational experience is reported for some real-life problems.

Mathematical Programming and Graph-Based Tools in Process Systems Engineering

Jiří Jaromír Klemeš, Brno University of Technology, Brno, Czech Republic; Pázmány Péter Catholic University, Budapest, Hungary

Abstract

Process Systems Engineering have traditionally used Mathematical Programming (MP) with creating a superstructure and using various optimisation methods to obtain an optimum. MP methods enable the search among many design alternatives and explicitly account for both investment and operating costs. MP problems may be generally formulated as Mixed Integer Nonlinear Programming (MINLP) problems. However, to avoid local optima, in many cases, the problem needs to be simplified to reformulate MINLP into Linear Programming (LP) problem, Nonlinear Programming (NLP) problem or Mixed Integer Linear Programming (MILP). Some efficient approaches have been presented in the past as MILP-NLP or MILP-MINLP.

This methodology has been well rehearsed over the years, with escalating available computing power meaning the main issue of the computing time being overcome. However, there are still key issues to be dealt with – such as local optima of MINLP (Mixed Integer Nonlinear Programming) problems and especially exploiting physical insights and industrial experience during the solution development. However, as can be seen from the example of Pinch Analysis, applying physical insight helps reduce the search space and steer solutions towards the global optimum.

A big step forward has been brought about by P-graph, a methodology using graphs combined with powerful optimisation. This has been a lean, streamlined tool for Process Network Synthesis, and despite a strong MP lobby become more and more spreading out.

However, as engineers, especially practising engineers in the industry, are traditionally by nature preferring the graphical insight for understanding processes as well as the interpretation of results, some other methodologies have been developed.

The traditional Process Integration based on Pinch Methodology tools to analyse and target process performance, starting firstly with Heat Exchanger Network (HEN), facilitated the development, which included Composite Curves, Grand Composite Curves, Time Slices and Time Average Composite Curves and classical Grid Diagram. These tools have been extended to Total Sites (integration of integrated processes) and further to Local Energy Integrated Sectors (covering beside industrial units also various civic, business, cities and even agriculture units). For visualisation, analysis and optimisation of those sites have been developed Site Source – Sink Profiles, Site Utilities Grand Composite Curves and Exergy Site Profiles. Extensions to resource optimisation and targeting has led to the introduction of Water and Waste Water Profiles, Power Composite Curves, Power Site Profiles, Emissions Composite Curves and a number of the others. However those tools have been also supported with developed numerical tools, including Total Site Problem Table Algorithm (TS-PTA), Segregated Problem Table Algorithm (SePTA), Total Site Sensitivity Table (TSST), Total Site Utility Distribution, Time Super Targeting and some others.

The very recent developments have resulted in (i) the Energy Transfer Diagram, (ii) Heat-Exchanger Load Diagram, (iii) Heat Surplus-Deficit Table, (iv) Shifted Retrofit Thermodynamic Diagram, (v) Retrofit Tracing Grid Diagram, (vi) Stream Temperature vs. Enthalpy Plot – STEP, and (vii) Retrofit Dashboard. Each of these visualisation tools provides a lens to increase the understanding of existing process. Using this insight together with defined steps and rules, it becomes easier to design and implement effective grassroots design and especially retrofit strategies which are at the cornerstone of PSE.

This lecture has been an attempt to present an overview of mainly graph-based approaches to raise awareness of usefulness of graphical representation-based method as well as an exploitation the hybrid approach.

Optimization and behavior

Ulrike Leopold-Wildburger, University of Graz, Graz, Austria

Abstract

While methods of OR represent the field of a science for delivering better decisions using optimal (or near-optimal) solutions to complex decision-making problems our actual behavior in practical applications quite often has to deal with non-fully rational decision makers. We try to make aware the tension between the two scopes and we will support this research by some examples. Coming from the field of OR we are aware that that techniques such as mathematical modeling, statistical analysis, and mathematical optimization are engaged in applications of advanced analytical methods with the aim to make better decisions. However, in everyday life OR is not executed in its pure version but often connected with other fields and disciplines, as psychology and behavioral sciences, microeconomics and even nowadays integrating neuroscience.

Some characteristic examples from the field of game theory will be prepared and checked with the actual behavior of decision makers in specific economic situations. We will deal with topics as cooperation, fairness and honesty and we will try to compare theoretical concepts with empirical data. 

Risk-Averse Control of Markov System

Andrzej Ruszczynski, Rutgers University, Piscataway NJ, USA

Abstract

We shall focus on modeling risk in dynamical systems and discuss fundamental properties of dynamic measures of risk. Special attention will be paid to the local property and the property of time consistency. Then we shall focus on risk-averse control of discrete-time Markov systems. We shall introduce the class of Markovian risk measures, and derive their structure. This will allow us to derive a risk-averse couterpart of dynamic programming equations. Then we shall extend these ideas to partially-observable systems and continuous-time Markov chains and derive the structure of risk measures and dynamic programming equations in these cases as well. In the last part of the talk, we shall discuss risk-averse control of diffusion processes and present a risk-averse counterpart of the Hamilton--Jacobi--Bellman equation.

Optimization challenges in Kidney Exchange Programs: past, present and future

Ana Viana, Centre for Industrial Engineering and Management of INESC TEC, Porto, Portugal

Abstract

Kidney exchange programmes (KEPs) represent an alternative of transplant for patients with an incompatible living donor: in these programs patients with a willing incompatible donor join a pool of incompatible patient-donor pairs and, if compatibility between patient in one pair and donor in another is found, patient in one pair can receive an organ from the donor in another pair and vice-versa. Pair-matching should be done in such a way that maximum social welfare (that can be measured by e.g. the number of transplants that will be performed) is achieved.

Pioneering contributions in this area used integer programming as a natural framework to represent the problem and find an optimal solution. However, those models did not consider data uncertainty, associated to pair dropout or undetected incompatibilities, which in practice may have a considerable impact on the solution to implement. Given that, current research has been addressing data uncertainty in different manners: robust optimization, stochastic optimization, and simulation.

In this talk we will present different Integer Programming models proposed in the literature for KEP, when data is considered to be certain. We will proceed with the description of some approaches that address data uncertainty. Finally, we provide some insights on future challenges in the area. In particular, we focus on the case where multiple agents collaborate in a joint pool.