MidTermReview - Stochastic Programming and Financial...

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Unformatted text preview: Stochastic Programming and Financial Analysis IE447 Midterm Review Dr. Ted Ralphs 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 ) ≤ 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? 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. 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.” 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 ≥ ∀ j ∈ N 1 x j ≤ ∀ 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. 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. 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. 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 ....
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This note was uploaded on 02/29/2008 for the course IE 447 taught by Professor Ralphs during the Spring '08 term at Lehigh University .

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MidTermReview - Stochastic Programming and Financial...

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