Surrogate modeling for geometry optimization in material design

We propose a new approach based on surrogate modeling for geometry optimization in material design. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed-integer models for topology optimization in sheet metal design

Topology optimization lies at the heart of many design tasks in mechanical engineering. For those sheet metal products that consist of a bundle of separate channels (such as conduits) we formulate the design task as a linear mixed-integer optimization problem. The design goal is to find a topology where each channel has a given cross section area, using a minimum amount of sheet met al. In addition to a light-weight design, stiffness should also be taken into account. The entire approach is demonstrated in the design of a conduit with five separate channels. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Global optimization of multilevel electricity market models including network design and graph partitioning

We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in Grimm et al. (2019) that does not include a network design problem. The two classical discrete optimization problems of network design and graph partitioning together with nonlinearities due to economic modeling yield extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof including novel primal-dual bound tightening techniques, whereas the second is a tailored generalized Benders decomposition. For the latter, we strengthen the Benders cuts of Grimm et al. (2019) by using the structure of the newly introduced network design subproblem. We prove for both methods that they yield global optimal solutions. Afterward, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation. Finally, we present a case study that illustrates the economic effects that are captured in our model.     

Enhancing finite difference approximations for double barrier options: mesh optimization and repeated Richardson extrapolation

We show that the performances of the finite difference method for double barrier option pricing can be strongly enhanced by applying both a repeated Richardson extrapolation technique and a mesh optimization procedure. In particular, first we construct a space mesh that is uniform and aligned with the discontinuity points of the solution being sought. This is accomplished by means of a suitable transformation of coordinates, which involves some parameters that are implicitly defined and whose existence and uniqueness is theoretically established. Then, a finite difference scheme employing repeated Richardson extrapolation in both space and time is developed. The overall approach exhibits high efficacy: barrier option prices can be computed with accuracy close to the machine precision in less than one second. The numerical simulations also reveal that the improvement over existing methods is due to the combination of the mesh optimization and the repeated Richardson extrapolation.

     

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.     

Flow simulation and shape optimization for aircraft design

Within the framework of the German aerospace research program, the CFD project MEGADESIGN was initiated. The main goal of the project is the development of efficient numerical methods for shape design and optimization. In order to meet the requirements of industrial implementations a co-operative effort has been set up which involves the German aircraft industry, the DLR, several universities and some small enterprises specialized in numerical optimization. This paper outlines the planned activities within MEGADESIGN, the status at the beginning of the project and it presents some early results achieved in the project.     

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.     

DSLC-FOA : Improved fruit fly optimization algorithm for application to structural engineering design optimization problems

In this study, we propose an improved fruit fly optimization algorithm (FOA) based on linear diminishing step and logistic chaos mapping (named DSLC-FOA) for solving benchmark function unconstrained optimization problems and constrained structural engineering design optimization problems. Based on comparisons with genetic algorithm, particle swarm optimization, FOA, LGMS -FOA, and chaotic FOA methods, we demonstrated that DSLC-FOA performed better at searching for the optimal solutions of four typical benchmark functions. The approximate optimal results were obtained using DSLC-FOA for three structural engineering design optimization problems as examples of applications. The numerical results demonstrated that the proposed DSLC-FOA algorithm is superior to the basic FOA and other metaheuristic or deterministic methods.     

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.     

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...