MidTermReview - Stochastic Programming and Financial...

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Stochastic Programming and Financial Analysis IE447 Midterm Review Dr. Ted Ralphs
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IE447 Midterm Review 1 Forming a Mathematical Programming Model The general form of a mathematical programming model is: min f ( x ) s.t. g ( x ) 0 h ( x ) = 0 x X where f : R n R , g : R n R m , h : R n R l , and X may be a discrete set, such as Z n . Notes : There is an important assumption here that all input data are known and fixed . Such a mathematical program is called deterministic . Is this realistic?
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IE447 Midterm Review 2 Categorizing Mathematical Programs Deterministic mathematical programs can be categorized along several fundamental lines. Constrained vs. Unconstrained Convex vs. Nonconvex Linear vs. Nonlinear Discrete vs. Continuous What is the importance of these categorizations? Knowing what category an instance is in can tell us something about how difficult it will be to solve. Different solvers are designed for different categories.
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IE447 Midterm Review 3 Unconstrained Optimization When M = and X = R n , we have an unconstrained optimization problem . Unconstrained optimization problems will not generally arise directly from applications. They do, however, arise as subproblems when solving mathematical programs. In unconstrained optimization, it is important to distinguish between the convex and nonconvex cases. Recall that in the convex case, optimizing globally is “easy.”
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IE447 Midterm Review 4 Linear Programs A linear program is one that can be written in a form in which the functions f and g i , i M are all linear and X = R n . In general, a linear program is one that can be written as minimize c x s . t . a i x b i i M 1 a i x b i i M 2 a i x = b i i M 3 x j 0 j N 1 x j 0 j N 2 Equivalently, a linear program can be written as minimize c x s . t . Ax b Generally speaking, linear programs are also “easy” to solve.
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IE447 Midterm Review 5 Nonlinear Programs A nonlinear program is any mathematical program that cannot be expressed as a linear program. Usually, this terminology also assumes X = R n . Note that by this definition, it is not always obvious whether a given instance is really nonlinear. In general, nonlinear programs are difficult to solve to global optimality.
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IE447 Midterm Review 6 Special Case: Convex Programs A convex program is a nonlinear program in which the objective function f is convex and the feasible region is a convex set. In practice, convex programs are usually “easy” to solve.
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IE447 Midterm Review 7 Special Case: Quadratic Programs If all of the functions f and g i for i M are quadratic functions, then we have a quadratic program . Often, the term quadratic program refers specifically to a program of the form minimize 1 2 x Qx + c x s . t . Ax b Because x Qx = 1 2 x ( Q + Q ) x , we can assume without loss of generality that Q is symmetric .
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