Information about Robust Optimization
In mathematics, robust optimization is an approach in optimization to deal with uncertainty. It is similar to the recourse model of stochastic programming, in that some of the parameters are random variables, except that feasibility for all possible realizations (called scenarios) is replaced by a penalty function in the objective. As such, the approach integrates goal programming with a scenario-based description of problem data. To illustrate, consider the LP:
where d, B, C and e are random variables with possible realizations , where N = the number of scenarios. The robust optimization model for this LP is:
where f is a function that measures the cost of the policy, P is a penalty function, and w > 0 (a parameter to trade off the cost of infeasibility). One example of f is the expected value:
, where p(s) = probability of scenario s. In a worst-case model, 
. The penalty function is defined to be zero if (x, y) is feasible (for all scenarios) -- i.e., P(0)=0. In addition, P satisfies a form of monotonicity: worse violations incur greater penalty. This often has the form 
-- i.e., the "up" and "down" penalties, where U and V are strictly increasing functions.
The above makes robust optimization similar (at least in the model) to a goal program. Recently, the robust optimization community defines it differently – it optimizes for the worst-case scenario. Let the uncertain MP be given by
where d, B, C and e are random variables with possible realizations , where N = the number of scenarios. The robust optimization model for this LP is:
and 
where f is a function that measures the cost of the policy, P is a penalty function, and w > 0 (a parameter to trade off the cost of infeasibility). One example of f is the expected value:
, where p(s) = probability of scenario s. In a worst-case model, . The penalty function is defined to be zero if (x, y) is feasible (for all scenarios) -- i.e., P(0)=0. In addition, P satisfies a form of monotonicity: worse violations incur greater penalty. This often has the form -- i.e., the "up" and "down" penalties, where U and V are strictly increasing functions.
The above makes robust optimization similar (at least in the model) to a goal program. Recently, the robust optimization community defines it differently – it optimizes for the worst-case scenario. Let the uncertain MP be given by
References
H.J. Greenberg. Mathematical Programming Glossary. World Wide Web, [1] 1996-2006. Edited by the INFORMS Computing Society. In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set.
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Stochastic programming is a framework for modeling optimization problems that involve uncertainty. Whereas deterministic optimization problems are formulated with known parameters, real world problems almost invariably include some unknown parameters.
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A random variable is an abstraction of the intuitive concept of chance into the theoretical domains of mathematics, forming the foundations of probability theory and mathematical statistics.
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Penalty methods are a certain class of algorithms to solve constraint optimization problems.
The penalty method replaces a constraint optimization problem by a series of unconstrained problems whose solutions must converge to the solution of the original constrained problem.
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The penalty method replaces a constraint optimization problem by a series of unconstrained problems whose solutions must converge to the solution of the original constrained problem.
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Goal programming is a branch of multiobjective optimization, which in turn is a branch of multi-criteria decision analysis (MCDA), also known as multiple-criteria decision making (MCDM).
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Goal programming is a branch of multiobjective optimization, which in turn is a branch of multi-criteria decision analysis (MCDA), also known as multiple-criteria decision making (MCDM).
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