Integer linear programming models for topology optimization in sheet metal design

The process of designing new industrial products is in many cases solely based on the intuition and experience of the responsible design engineer. The aid of computers is restricted to visualization and manual manipulation tools. We demonstrate that the design process for conduits, which are made out of sheet metal plates, can be supported by mathematical optimization models and solution techniques, leading to challenging optimization problems. The design goal is to find a topology that consists of several channels with a given cross section area using a minimum amount of sheet metal and, at the same time, maximizing its stiffness. We consider a mixed integer linear programming model to describe the topology of two dimensional slices of a three dimensional sheet metal product. We give different model formulations, based on cuts and on multicommodity flows. Numerical results for various test instances are presented.     

Electricity swing options: Behavioral models and pricing

Electricity swing options are supply contracts for power, which give the owner the right to change the required delivery on short time notice. It gives more flexibility than fixed base load or peak load contracts. The name “option” is a bit misleading, since it gives the owner multiple exercise rights at many different time horizons with exercise amounts on a continuous scale. We look at the problem to determine a rational ask price for such a contract from the viewpoint of the contract seller. The pricing of these contracts differs drastically from the pricing of financial options. First, peculiar properties arise from the non-storability of the underlying (the energy) and therefore the impossibility to hedge with the underlying, hedging is only possible with some future contracts. Second, the behavior of the owner plays an important role. Based on some behavioral model for the option holder, we develop a game-theoretic model, which allows to identify the equilibrium price. Besides some theoretical results, we present some numerical results which clarify the dependence of the asked price on the amount of flexibility offered in the swing option.     

A structural framework for the design of integrated environmental and land-use planning optimization models

This study presents a structural framework for analyzing land-use/environmental interactions and formulating planning models accounting for these interactions. A general conceptual planning model is first developed. Its applicability is illustrated through a review of major environmental pollution transfer models, and through the development of a prototypical model that is progressively expanded to account for centralized treatments, transfer modifications, short-term and long-term dynamics, and the stochasticity of the environment.     

Low-dimensional Yang-Mills theories: Matrix models and emergent geometry

In a simple example of a bosonic three-matrix model, we show how a background geometry can condense as the temperature or coupling constant passes through a critical value. We show that this example belongs to a new universality class of phase transitions where the background geometry is itself emergent.     

Communication protocols for options and results in a distributed optimization environment

Much has been written about optimization instance formats. The MPS standard for linear mixed-integer programs is well known and has been around for many years. Other extensible formats are available for other optimization categories such as stochastic and nonlinear programming. However, the problem instance is not the only piece of information shared between the instance generator and the solver. Solver options and solver results must also be communicated. To our knowledge there is no commonly accepted format for representing either solver options or solver results. In this paper we propose a framework and theory for solver option and solver result representation in a modern distributed computing environment. A software implementation of the framework is available as an open-source COIN-OR project.     

A design and a geometry for the group Fi 22

The Fischer group Fi ₂₂ acts as a rank 3 group of automorphisms of a symmetric 2-(14080,1444,148) design. This design does not have a doubly transitive automorphism group, since there is a partial linear space with lines of size 4 defined combinatorially from the design and preserved by its automorphism group. We investigate this geometry and determine the structure of various subspaces of it.      

Matrix models, complex geometry, and integrable systems: II

We consider certain examples of applications of the general methods based on geometry and integrability of matrix models. These methods were described in the first part of this paper. In particular, we investigate the nonlinear differential equations satisfied by semiclassical tau functions. We also discuss a similar semiclassical geometric picture arising in the context of multidimensional supersymmetric gauge theories and the AdS/CFT correspondence. [This article was written at the request of the Editorial Board. It is based on several lectures presented at schools of mathematical physics and talks at the conference “Complex Geometry and String Theory” and the Polivanov memorial seminar.] __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 399–449, June, 2006.     

Matrix models, complex geometry, and integrable systems: I

We consider the simplest gauge theories given by one-and two-matrix integrals and concentrate on their stringy and geometric properties. We recall the general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. We study the main ingredients of this geometric picture, suggesting that it can be generalized beyond one complex dimension, and formulate them in terms of semiclassical integrable systems solved by constructing tau functions or prepotentials. We discuss the complex curves and tau functions of one-and two-matrix models in detail. [This article was written at the request of the Editorial Board. It is based on several lectures presented at schools of mathematical physics and talks at the conference “Complex Geometry and String Theory” and the Polivanov memorial seminar.] __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 163–228, May, 2006.     

Reduced quasi-Newton method for simultaneous design and optimization

We consider the task of design optimization where the constraint is a state equation that can only be solved by a typically rather slowly converging fixed point solver. This process can be augmented by a corresponding adjoint solver and based on the resulting approximate reduced derivatives also an optimization iteration which actually changes the design. To coordinate the three iterative processes, we use an exact penalty function of doubly augmented Lagrangian type. The main issue here is how to derive a design space preconditioner for the approximated reduced gradient which ensures a consistent reduction of the employed penalty function as well as significant design corrections. Some numerical experiments for an alternating approach where any combination and sequencing of steps are used to improve feasibility and optimality done on a variant of the Bratu problem are presented.     

Structuring resource allocation decisions: A framework for building multi-criteria portfolio models with area-grouped options

Multi-criteria portfolio modelling has been extensively employed as an effective means to allocate scarce resources for investment in projects when considering costs, benefits and risks. Some of these modelling approaches allow the grouping of projects into organisational areas, thus also supporting the decision of resource allocation among organisational units in a way that is collectively efficient for the organisation. However, structuring in practice a portfolio model using this latter type of approach is not a trivial task. How should areas be defined? Where should new projects be included? How should one define the criteria to evaluate performance? As far as we know, there is very little indication in the operational research and decision sciences literatures on how to structure this type of model. This paper suggests different ways to structuring portfolio models where projects are divided into areas and evaluated by multiple criteria, and illustrates their use in two action-research projects. Drawing on these experiences it then suggests a general framework for the structuring of such models in practice. Directions for future research are also identified.     

Global optimization method using adaptive and parallel ensemble of surrogates for engineering design optimization

As a robust and efficient technique for global optimization, surrogate-based optimization method has been widely used in dealing with the complicated and computation-intensive engineering design optimization problems. It’s hard to select an appropriate surrogate model without knowing the behaviour of the real system a priori in most cases. To overcome this difficulty, a global optimization method using an adaptive and parallel ensemble of surrogates combining three representative surrogate models with optimized weight factors has been proposed. The selection of weight factors is treated as an optimization problem with the desired solution being one that minimizes the generalized mean square cross-validation error. The proposed optimization method is tested by considering several well-known numerical examples and one industrial problem compared with other optimization methods. The results show that the proposed optimization method can be a robust and efficient approach in surrogate-based optimization for locating the global optimum.     

An alternative to EM for Gaussian mixture models: batch and stochastic Riemannian optimization

Mathematical Programming - We consider maximum likelihood estimation for Gaussian Mixture Models (Gmm s). This task is almost invariably solved (in theory and practice) via the Expectation...     

Matrix monotonicity and self-concordance: how to handle quantum entropy in optimization problems

Let g be a continuously differentiable function whose derivative is matrix monotone on the positive semi-axis. Such a function induces a function \(\varphi (x)=\mathrm{tr}\, \big (g(x)\big )\) on the cone of squares of an arbitrary Euclidean Jordan algebra. We show that \(\varphi (x) - \ln \det (x)\) is a self-concordant function on the interior of the cone. We also show that \( -\ln (t-\varphi (x))-\ln \det (x)\) is \((r+1)\)-self-concordant barrier on the epigraph of \(\varphi ,\) where r is the rank of the Jordan algebra. The case \(\phi (x)=\mathrm{tr(x\ln x)}\) is discussed in detail.     

Covering models and optimization techniques for emergency response facility location and planning: a review

With emergencies being, unfortunately, part of our lives, it is crucial to efficiently plan and allocate emergency response facilities that deliver effective and timely relief to people most in need. Emergency Medical Services (EMS) allocation problems deal with locating EMS facilities among potential sites to provide efficient and effective services over a wide area with spatially distributed demands. It is often problematic due to the intrinsic complexity of these problems. This paper reviews covering models and optimization techniques for emergency response facility location and planning in the literature from the past few decades, while emphasizing recent developments. We introduce several typical covering models and their extensions ordered from simple to complex, including Location Set Covering Problem (LSCP), Maximal Covering Location Problem (MCLP), Double Standard Model (DSM), Maximum Expected Covering Location Problem (MEXCLP), and Maximum Availability Location Problem (MALP) models. In addition, recent developments on hypercube queuing models, dynamic allocation models, gradual covering models, and cooperative covering models are also presented in this paper. The corresponding optimization techniques to solve these models, including heuristic algorithms, simulation, and exact methods, are summarized.     

On similarities between two models of global optimization: statistical models and radial basis functions

Construction of global optimization algorithms using statistical models and radial basis function models is discussed. A new method of data smoothing using radial basis function and least squares approach is presented. It is shown that the P-algorithm for global optimization in the presence of noise based on a statistical model coincides with the corresponding radial basis algorithm.     

Estimation and model selection in a class of semiparametric models for cluster data

Stimulated by a study in Bangladesh about the first birth interval, we propose a semivarying-coefficient model for cluster data analysis. We consider the estimation procedure for the proposed model and establish the asymptotic results of the proposed estimators. Furthermore, we employ the cross-validation (CV) to identify the constant coefficients. The associated asymptotic properties are rigorously examined. Simulation studies are conducted to investigate the performance of the proposed estimation and the CV-based model selection procedure for finite sample size. Finally, our methods are used to analyse the aforementioned data set to explore how several factors affect the first birth interval in Bangladesh.     

Asymmetric risk measures and tracking models for portfolio optimization under uncertainty

Traditional asset allocation of the Markowitz type defines risk to be the variance of the return, contradicting the common-sense intuition that higher returns should be preferred to lower. An argument of Levy and Markowitz justifies the mean/variance selection criteria by deriving it from a local quadratic approximation to utility functions. We extend the Levy-Markowitz argument to account for asymmetric risk by basing the local approximation onpiecewise linear-quadratic risk measures, which can be tuned to express a wide range of preferences and adjusted to reject outliers in the data. The implications of this argument lead us to reject the commonly proposed asymmetric alternatives, the mean/lower partial moment efficient frontiers, in favor of the risk tolerance frontier. An alternative model that allows for asymmetry is the tracking model, where a portfolio is sought to reproduce a (possibly) asymmetric distribution at lowest cost.