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## Bilevel derivative-free optimization and its application to robust optimization

Fonte: Centro de Matemática da Universidade de Coimbra
Publicador: Centro de Matemática da Universidade de Coimbra

Tipo: Pré-impressão

ENG

Relevância na Pesquisa

66.33%

#Bilevel programming#Derivative-free optimization#Robust optimization#Simulation-based optimization#Trust-region methods#Quadratic interpolation

We address bilevel programming problems when the derivatives of both
the upper and the lower level objective functions are unavailable.
The core algorithms used for both levels are trust-region interpolation-based
methods, using minimum Frobenius norm quadratic models when the number of
points is smaller than the number of basis components. We take advantage of the
problem structure to derive conditions (related to the global convergence theory of
the underlying trust-region methods, as far as possible) under which the lower level
can be solved inexactly and sample points can be reused for model building. In addition,
we indicate numerically how effective these expedients can be. A number of
other issues are also discussed, from the extension to linearly constrained problems
to the use of surrogate models for the lower level response.
One important application of our work appears in the robust optimization of
simulation-based functions, which may arise due to implementation variables or
uncertain parameters. The robust counterpart of an optimization problem without
derivatives falls in the category of the bilevel problems under consideration here.
We provide numerical illustrations of the application of our algorithmic framework
to such robust optimization examples

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## On the robustness of structural risk optimization with respect to epistemic uncertainties.

Fonte: Redding
Publicador: Redding

Tipo: Artigo de Revista Científica

ENG

Relevância na Pesquisa

56.39%

#risk analysis#representation of uncertainty#robust optimization#structural reliability#fuzzy variables#epistemic uncertainty

In the context of structural design, risk optimization allows one to find a proper point of balance between the concurrent goals of economy and safety. Risk optimization involves the minimization of total expected costs, which include expected costs of failure. Expected costs of failure are evaluated from nominal failure probabilities, which reflect the analystï¿½s degree of belief in the structureï¿½s performance. Such failure probabilities are said to be nominal because they are evaluated from imperfect and/or incomplete mechanical, mathematical and probabilistic models. Hence, model uncertainty and other types of epistemic uncertainties are likely to compromise the results of risk optimization. In this paper, the concept of robustness is employed in order to find risk optimization solutions which are less sensitive to epistemic uncertainties. The investigation is based on a simple but illustrative problem, which is built from an elementary but fundamental structural (load-resistance) reliability problem. Intrinsic or aleatoric uncertainties, which can be quantified probabilistically and modeled as random variables or stochastic processes, are incorporated in the underlying structural reliability problem. Epistemic uncertainties that can only be quantified possibilistically are modeled as fuzzy variables...

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## Programação estocástica e otimização robusta no planejamento da produção de empresas moveleiras; Stochastic programming and robust optimization in the production planning of furniture industries

Fonte: Biblioteca Digitais de Teses e Dissertações da USP
Publicador: Biblioteca Digitais de Teses e Dissertações da USP

Tipo: Tese de Doutorado
Formato: application/pdf

Publicado em 08/04/2011
PT

Relevância na Pesquisa

66.34%

#Combined lot-sizing and cutting-stock problem#Empresas moveleiras#Furniture industries#Otimização robusta#Problemacombinado de dimensionamento de lotes e corte de estoque#Programação estoástica#Robust optimization#Stochastic programming

O planejamento da produção em indústrias moveleiras de pequeno porte é comumente constituído por decisões referentes ao volume de produção e à política de estoque, com o objetivo de minimizar o desperdício de material, os atrasos e as horas-extras utilizadas ao longo do horizonte de planejamento. Administrar tais decisões de uma maneira tratável e eficiente é, em geral, um desafio, especialmente considerando a natureza incerta dos dados. Nessa tese, são desenvolvidos modelos de otimização para apoiar tais decisões no contexto do problema combinado de dimensionamento de lotes e corte de estoque sob incertezas que surge em indústrias moveleiras. Para lidar com as incertezas dos dados, são investigadas duas metodologias: programação estocástica e otimização robusta. Dessa maneira, são propostos modelos de programação estocástica de dois estágios com recurso, assim como modelos estocásticos robustos que incorporam aversão ao risco. A motivação em também desenvolver modelos baseados em otimização robusta é considerar casos práticos em que não há uma descrição probabilística explícita dos dados de entrada, assim como evitar trabalhar com numerosos cenários, o que pode tornar o modelo estocástico computacionalmente intratável. Os experimentos numéricos baseados em exemplares reais de uma empresa moveleira de pequeno porte mostram que as soluções obtidas pelos modelos de programação estocástica fornecem planos de produção robustos e que o (a) decisor (a) pode designar suas preferências em relação ao risco aos modelos...

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## Otimização linear robusta multitemporal de uma carteira de ativos com parâmetros de média e dispersão incertos; Robust linear multistage portfolio optimization with location and dispersion parameters subject to uncertainty.

Fonte: Biblioteca Digitais de Teses e Dissertações da USP
Publicador: Biblioteca Digitais de Teses e Dissertações da USP

Tipo: Tese de Doutorado
Formato: application/pdf

Publicado em 27/09/2011
PT

Relevância na Pesquisa

56.29%

#Administração de carteiras#Asset allocation#D-Norm#Linear programming#Múltilplos estágios#Multiple periods#Norma D#Otimização robusta#Programação linear#Robust optimization

Nos últimos anos, percebeu-se um avanço substancial das metodologias sistemáticas de seleção de ativos em portfólios financeiros, baseadas em técnicas de otimização. A maior pressão por desempenho sobre as gestoras de recursos e a evolução dos softwares e pacotes de otimização foram fatores que contribuíram para esse desenvolvimento. Dentre as técnicas mais reconhecidas utilizadas na gestão de portfólios está a de otimização robusta, cuja aplicação na solução de problemas com dados incertos iniciou-se na década de 1970 e, desde então, vem evoluindo em sofisticação. Partindo de uma extensão recente do método, propõe-se um novo modelo linear que resolve o problema de otimização de um portfólio para múltiplos estágios, com inovações no tratamento da incerteza das estimativas de dispersão dos retornos. Os resultados mostram que o método proposto desempenha muito bem em termos de rentabilidade e de métricas de risco-retorno em momentos de turbulência dos mercados. Por fim, demonstra-se empiricamente que o modelo alcança um desempenho ainda melhor em termos de rentabilidade com a adoção de um estimador eficiente para o valor esperado dos retornos e com a simultânea redução do nível de robustez do modelo.; It has been realized in the last years a remarkable development of the optimization techniques to solve the problem of financial portfolio selection. The pressure on asset management firms to maintain a more stable performance and the evolution of specialized software packages have enabled this positive trend. One of the most recognized approaches applied to the management of investments is the robust optimization...

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## Otimização robusta aplicada à operação de reservatórios para a geração de energia.; Robust optimization applied to reservoirs operation for hydropower generation.

Fonte: Biblioteca Digitais de Teses e Dissertações da USP
Publicador: Biblioteca Digitais de Teses e Dissertações da USP

Tipo: Dissertação de Mestrado
Formato: application/pdf

Publicado em 02/07/2013
PT

Relevância na Pesquisa

66.25%

#Geração de energia#Otimização robusta#Power generation#Reservatórios#Reservoirs#Robust optimization

Este trabalho tem como objetivo avaliar a viabilidade da aplicação de técnicas de otimização robusta (OR) no planejamento da operação de reservatórios para geração de energia hidrelétrica. A OR é uma técnica de otimização que visa encontrar resultados que sejam menos sensíveis às incertezas nas variáveis do modelo através da minimização da variância da função objetivo para diferentes cenários. Desta forma foi desenvolvido um modelo de otimização robusta aplicado à operação de reservatórios para a geração de energia hidrelétrica, chamado HIDRO-OR, utilizando o software General Algebraic Modeling System (GAMS). Para estudo de caso foram utilizados os dados da UHE Sinop, a ser instalada no rio Teles Pires MT. Inicialmente foi realizada uma análise de sensibilidade utilizando diferentes combinações dos coeficientes de ponderação da função objetivo e três conjuntos de cenários. Nesta abordagem, o modelo resultou em vertimentos indesejados para realizar a diminuição do desvio padrão dos resultados entre os diferentes cenários. Uma solução encontrada para o problema foi realizar a otimização em duas etapas. Na primeira etapa ocorre a otimização robusta propriamente dita e são fixados os resultados para o primeiro mês de operação. Na segunda etapa...

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## A Robust Optimization Approach to Supply Chain Management

Fonte: MIT - Massachusetts Institute of Technology
Publicador: MIT - Massachusetts Institute of Technology

Tipo: Artigo de Revista Científica
Formato: 413796 bytes; application/pdf

EN_US

Relevância na Pesquisa

66.13%

We propose a general methodology based on robust optimization to address the problem of optimally controlling a supply chain subject to stochastic demand in discrete time. The attractive features of the proposed approach are: (a) It incorporates a wide variety of phenomena, including demands that are not identically distributed over time and capacity on the echelons and links; (b) it uses very little information on the demand distributions; (c) it leads to qualititatively similar optimal policies (basestock policies) as in dynamic programming; (d) it is numerically tractable for large scale supply chain problems even in networks, where dynamic programming methods face serious dimensionality problems; (e) in preliminary computation experiments, it often outperforms dynamic programming based solutions for a wide range of parameters.; Singapore-MIT Alliance (SMA)

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## Robust fluid control of multiclass queueing networks; Robust fluid control of multiclass queuing networks

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 92 p.

ENG

Relevância na Pesquisa

56.34%

This thesis applies recent advances in the field of robust optimization to the optimal control of multiclass queueing networks. We develop models that take into account the uncertainty of interarrival and service time in multiclass queueing network problems without assuming a specific probability distribution, while remaining highly tractable and providing insight into the corresponding optimal control policy. Our approach also allows us to adjust the level of robustness of the solution to trade off performance and protection against uncertainty. We apply robust optimization to both open and closed queueing networks. For open queueing networks, we study control problems that involve sequencing, routing and input control decision, and optimize the total holding cost. For closed queueing networks, we focus on the sequencing problem and optimize the throughput. We compare the robust solutions to those derived by fluid control, dynamic programming and stochastic input control. We show that the robust control policy leads to better performance. Robust optimization emerges as a promising methodology to address a wide range of multiclass queueing networks subject to uncertainty, as it leads to representations of randomness that make few assumptions on the underlying probabilities. It also remains numerically tractable...

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## Empirical comparison of robust, data driven and stochastic optimization

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 49 leaves

ENG

Relevância na Pesquisa

56.26%

In this thesis, we compare computationally four methods for solving optimization problems under uncertainty: * Robust Optimization (RO) * Adaptive Robust Optimization (ARO) * Data Driven Optimization (DDO) * stochastic Programming (SP) We have implemented several computation experiments to demonstrate the different performance of these methods. We conclude that ARO outperform RO, which has a comparable performance with DDO. SP has a comparable performance with RO when the assumed distribution is the same as the true underlying distribution, but under performs RO when the assumed distribution is different from the true distribution.; by Wang, Yanbo.; Includes bibliographical references (leaf 49).; Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008.

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## Likelihood Robust Optimization for Data-driven Problems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

56.3%

We consider optimal decision-making problems in an uncertain environment. In
particular, we consider the case in which the distribution of the input is
unknown, yet there is abundant historical data drawn from the distribution. In
this paper, we propose a new type of distributionally robust optimization model
called the likelihood robust optimization (LRO) model for this class of
problems. In contrast to previous work on distributionally robust optimization
that focuses on certain parameters (e.g., mean, variance, etc.) of the input
distribution, we exploit the historical data and define the accessible
distribution set to contain only those distributions that make the observed
data achieve a certain level of likelihood. Then we formulate the targeting
problem as one of optimizing the expected value of the objective function under
the worst-case distribution in that set. Our model avoids the
over-conservativeness of some prior robust approaches by ruling out unrealistic
distributions while maintaining robustness of the solution for any
statistically likely outcomes. We present statistical analyses of our model
using Bayesian statistics and empirical likelihood theory. Specifically, we
prove the asymptotic behavior of our distribution set and establish the
relationship between our model and other distributionally robust models. To
test the performance of our model...

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## Algorithm Engineering in Robust Optimization

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

56.3%

Robust optimization is a young and emerging field of research having received
a considerable increase of interest over the last decade. In this paper, we
argue that the the algorithm engineering methodology fits very well to the
field of robust optimization and yields a rewarding new perspective on both the
current state of research and open research directions.
To this end we go through the algorithm engineering cycle of design and
analysis of concepts, development and implementation of algorithms, and
theoretical and experimental evaluation. We show that many ideas of algorithm
engineering have already been applied in publications on robust optimization.
Most work on robust optimization is devoted to analysis of the concepts and the
development of algorithms, some papers deal with the evaluation of a particular
concept in case studies, and work on comparison of concepts just starts. What
is still a drawback in many papers on robustness is the missing link to include
the results of the experiments again in the design.

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## Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/05/2015

Relevância na Pesquisa

56.23%

We consider stochastic programs where the distribution of the uncertain
parameters is only observable through a finite training dataset. Using the
Wasserstein metric, we construct a ball in the space of (multivariate and
non-discrete) probability distributions centered at the uniform distribution on
the training samples, and we seek decisions that perform best in view of the
worst-case distribution within this Wasserstein ball. The state-of-the-art
methods for solving the resulting distributionally robust optimization problems
rely on global optimization techniques, which quickly become computationally
excruciating. In this paper we demonstrate that, under mild assumptions, the
distributionally robust optimization problems over Wasserstein balls can in
fact be reformulated as finite convex programs---in many interesting cases even
as tractable linear programs. Leveraging recent measure concentration results,
we also show that their solutions enjoy powerful finite-sample performance
guarantees. Our theoretical results are exemplified in mean-risk portfolio
optimization as well as uncertainty quantification.; Comment: 32 pages, 6 figures

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## Commitment and Dispatch of Heat and Power Units via Affinely Adjustable Robust Optimization

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/07/2015

Relevância na Pesquisa

56.3%

The joint management of heat and power systems is believed to be key to the
integration of renewables into energy systems with a large penetration of
district heating. Determining the day-ahead unit commitment and production
schedules for these systems is an optimization problem subject to uncertainty
stemming from the unpredictability of demand and prices for heat and
electricity. Furthermore, owing to the dynamic features of production and heat
storage units as well as to the length and granularity of the optimization
horizon (e.g., one whole day with hourly resolution), this problem is in
essence a multi-stage one. We propose a formulation based on robust
optimization where recourse decisions are approximated as linear or
piecewise-linear functions of the uncertain parameters. This approach allows
for a rigorous modeling of the uncertainty in multi-stage decision-making
without compromising computational tractability. We perform an extensive
numerical study based on data from the Copenhagen area in Denmark, which
highlights important features of the proposed model. Firstly, we illustrate
commitment and dispatch choices that increase conservativeness in the robust
optimization approach. Secondly, we appraise the gain obtained by switching
from linear to piecewise-linear decision rules within robust optimization.
Furthermore...

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## A cutting surface algorithm for semi-infinite convex programming with an application to moment robust optimization

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

56.4%

#Mathematics - Optimization and Control#Quantitative Finance - Computational Finance#Quantitative Finance - Portfolio Management#90C34, 90C15, 90C25, 90-08#G.1.6

We present and analyze a central cutting surface algorithm for general
semi-infinite convex optimization problems, and use it to develop a novel
algorithm for distributionally robust optimization problems in which the
uncertainty set consists of probability distributions with given bounds on
their moments. Moments of arbitrary order, as well as non-polynomial moments
can be included in the formulation. We show that this gives rise to a hierarchy
of optimization problems with decreasing levels of risk-aversion, with classic
robust optimization at one end of the spectrum, and stochastic programming at
the other. Although our primary motivation is to solve distributionally robust
optimization problems with moment uncertainty, the cutting surface method for
general semi-infinite convex programs is also of independent interest. The
proposed method is applicable to problems with non-differentiable semi-infinite
constraints indexed by an infinite-dimensional index set. Examples comparing
the cutting surface algorithm to the central cutting plane algorithm of
Kortanek and No demonstrate the potential of our algorithm even in the solution
of traditional semi-infinite convex programming problems whose constraints are
differentiable and are indexed by an index set of low dimension. After the rate
of convergence analysis of the cutting surface algorithm...

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## Adaptive Robust Optimization with Dynamic Uncertainty Sets for Multi-Period Economic Dispatch under Significant Wind

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

56.28%

The exceptional benefits of wind power as an environmentally responsible
renewable energy resource have led to an increasing penetration of wind energy
in today's power systems. This trend has started to reshape the paradigms of
power system operations, as dealing with uncertainty caused by the highly
intermittent and uncertain wind power becomes a significant issue. Motivated by
this, we present a new framework using adaptive robust optimization for the
economic dispatch of power systems with high level of wind penetration. In
particular, we propose an adaptive robust optimization model for multi-period
economic dispatch, and introduce the concept of dynamic uncertainty sets and
methods to construct such sets to model temporal and spatial correlations of
uncertainty. We also develop a simulation platform which combines the proposed
robust economic dispatch model with statistical prediction tools in a rolling
horizon framework. We have conducted extensive computational experiments on
this platform using real wind data. The results are promising and demonstrate
the benefits of our approach in terms of cost and reliability over existing
robust optimization models as well as recent look-ahead dispatch models.; Comment: Accepted for publication at IEEE Transactions on Power Systems

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## Data-Driven Robust Optimization

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

56.25%

The last decade witnessed an explosion in the availability of data for
operations research applications. Motivated by this growing availability, we
propose a novel schema for utilizing data to design uncertainty sets for robust
optimization using statistical hypothesis tests. The approach is flexible and
widely applicable, and robust optimization problems built from our new sets are
computationally tractable, both theoretically and practically. Furthermore,
optimal solutions to these problems enjoy a strong, finite-sample probabilistic
guarantee. \edit{We describe concrete procedures for choosing an appropriate
set for a given application and applying our approach to multiple uncertain
constraints. Computational evidence in portfolio management and queuing confirm
that our data-driven sets significantly outperform traditional robust
optimization techniques whenever data is available.; Comment: 38 pages, 15 page appendix, 7 figures. This version updated as of
Oct. 2014

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## Robust optimization with incremental recourse

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/12/2013

Relevância na Pesquisa

56.34%

In this paper, we consider an adaptive approach to address optimization
problems with uncertain cost parameters. Here, the decision maker selects an
initial decision, observes the realization of the uncertain cost parameters,
and then is permitted to modify the initial decision. We treat the uncertainty
using the framework of robust optimization in which uncertain parameters lie
within a given set. The decision maker optimizes so as to develop the best cost
guarantee in terms of the worst-case analysis. The recourse decision is
``incremental"; that is, the decision maker is permitted to change the initial
solution by a small fixed amount. We refer to the resulting problem as the
robust incremental problem. We study robust incremental variants of several
optimization problems. We show that the robust incremental counterpart of a
linear program is itself a linear program if the uncertainty set is polyhedral.
Hence, it is solvable in polynomial time. We establish the NP-hardness for
robust incremental linear programming for the case of a discrete uncertainty
set. We show that the robust incremental shortest path problem is NP-complete
when costs are chosen from a polyhedral uncertainty set, even in the case that
only one new arc may be added to the initial path. We also address the
complexity of several special cases of the robust incremental shortest path
problem and the robust incremental minimum spanning tree problem.

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## Oracle-Based Robust Optimization via Online Learning

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/02/2014

Relevância na Pesquisa

56.42%

Robust optimization is a common framework in optimization under uncertainty
when the problem parameters are not known, but it is rather known that the
parameters belong to some given uncertainty set. In the robust optimization
framework the problem solved is a min-max problem where a solution is judged
according to its performance on the worst possible realization of the
parameters. In many cases, a straightforward solution of the robust
optimization problem of a certain type requires solving an optimization problem
of a more complicated type, and in some cases even NP-hard. For example,
solving a robust conic quadratic program, such as those arising in robust SVM,
ellipsoidal uncertainty leads in general to a semidefinite program. In this
paper we develop a method for approximately solving a robust optimization
problem using tools from online convex optimization, where in every stage a
standard (non-robust) optimization program is solved. Our algorithms find an
approximate robust solution using a number of calls to an oracle that solves
the original (non-robust) problem that is inversely proportional to the square
of the target accuracy.

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## New results about multi-band uncertainty in Robust Optimization

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/08/2012

Relevância na Pesquisa

56.27%

#Mathematics - Optimization and Control#Computer Science - Data Structures and Algorithms#Computer Science - Numerical Analysis#90C05, 90C35, 90C57, 90C90

"The Price of Robustness" by Bertsimas and Sim represented a breakthrough in
the development of a tractable robust counterpart of Linear Programming
Problems. However, the central modeling assumption that the deviation band of
each uncertain parameter is single may be too limitative in practice:
experience indeed suggests that the deviations distribute also internally to
the single band, so that getting a higher resolution by partitioning the band
into multiple sub-bands seems advisable. The critical aim of our work is to
close the knowledge gap about the adoption of a multi-band uncertainty set in
Robust Optimization: a general definition and intensive theoretical study of a
multi-band model are actually still missing. Our new developments have been
also strongly inspired and encouraged by our industrial partners, which have
been interested in getting a better modeling of arbitrary distributions, built
on historical data of the uncertainty affecting the considered real-world
problems. In this paper, we study the robust counterpart of a Linear
Programming Problem with uncertain coefficient matrix, when a multi-band
uncertainty set is considered. We first show that the robust counterpart
corresponds to a compact LP formulation. Then we investigate the problem of
separating cuts imposing robustness and we show that the separation can be
efficiently operated by solving a min-cost flow problem. Finally...

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## A Practical Guide to Robust Optimization

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/01/2015

Relevância na Pesquisa

56.34%

Robust optimization is a young and active research field that has been mainly
developed in the last 15 years. Robust optimization is very useful for
practice, since it is tailored to the information at hand, and it leads to
computationally tractable formulations. It is therefore remarkable that
real-life applications of robust optimization are still lagging behind; there
is much more potential for real-life applications than has been exploited
hitherto. The aim of this paper is to help practitioners to understand robust
optimization and to successfully apply it in practice. We provide a brief
introduction to robust optimization, and also describe important do's and
don'ts for using it in practice. We use many small examples to illustrate our
discussions.

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## Multi-objective Robust Optimization using a Post-optimality Sensitivity Analysis Technique: Application to a Wind Turbine Design

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/12/2014

Relevância na Pesquisa

56.25%

Toward a multi-objective optimization robust problem, the variations in
design variables and design environment pa-rameters include the small
variations and the large varia-tions. The former have small effect on the
performance func-tions and/or the constraints, and the latter refer to the ones
that have large effect on the performance functions and/or the constraints. The
robustness of performance functions is discussed in this paper. A
post-optimality sensitivity analysis technique for multi-objective robust
optimization problems is discussed and two robustness indices are introduced.
The first one considers the robustness of the performance func-tions to small
variations in the design variables and the de-sign environment parameters. The
second robustness index characterizes the robustness of the performance
functions to large variations in the design environment parameters. It is based
on the ability of a solution to maintain a good Pareto ranking for different
design environment parameters due to large variations. The robustness of the
solutions is treated as vectors in the robustness function space, which is
defined by the two proposed robustness indices. As a result, the designer can
compare the robustness of all Pareto optimal solutions and make a decision.
Finally...

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