cs229-cvxopt2 - Convex Optimization Overview (cntd) Chuong...

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Convex Optimization Overview (cnt’d) Chuong B. Do November 29, 2009 During last week’s section, we began our study of convex optimization , the study of mathematical optimization problems of the form, minimize x R n f ( x ) subject to x C. (1) In a convex optimization problem, x R n is a vector known as the optimization variable , f : R n R is a convex function that we want to minimize, and C R n is a convex set describing the set of feasible solutions. From a computational perspective, convex optimiza- tion problems are interesting in the sense that any locally optimal solution will always be guaranteed to be globally optimal. Over the last several decades, general purpose methods for solving convex optimization problems have become increasingly reliable and e±cient. In these lecture notes, we continue our foray into the ²eld of convex optimization. In particular, we explore a powerful concept in convex optimization theory known as Lagrange duality . We focus on the main intuitions and mechanics of Lagrange duality; in particular, we describe the concept of the Lagrangian, its relation to primal and dual problems, and the role of the Karush-Kuhn-Tucker (KKT) conditions in providing necessary and su±cient conditions for optimality of a convex optimization problem. 1 Lagrange duality Generally speaking, the theory of Lagrange duality is the study of optimal solutions to convex optimization problems. As we saw previously in lecture, when minimizing a di³erentiable convex function f ( x ) with respect to x R n , a necessary and su±cient condition for x * R n to be globally optimal is that x f ( x * )= 0 . In the more general setting of convex optimization problem with constraints, however, this simple optimality condition does not work. One primary goal of duality theory is to characterize the optimal points of convex programs in a mathematically rigorous way. In these notes, we provide a brief introduction to Lagrange duality and its applications 1
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to generic diferentiable convex optimization problems oF the Form, minimize x R n f ( x ) subject to g i ( x ) 0 ,i =1 , . . . , m, h i ( x ) = 0 , . . . , p, (OPT) where x R n is the optimization variable , f : R n R and g i : R n R are diferen- tiable convex Functions 1 , and h i : R n R are a±ne Functions . 2 1.1 The Lagrangian In this section, we introduce an arti±cial-looking construct called the “Lagrangian” which is the basis oF Lagrange duality theory. Given a convex constrained minimization problem oF the Form (OPT), the (generalized) Lagrangian is a Function L : R n × R m × R p R , de±ned as L ( x, α, β )= f ( x )+ m ± i =1 α i g i ( x p ± i =1 β i h i ( x ) . (2) Here, the ±rst argument oF the Lagrangian is a vector x R n , whose dimensionality matches that oF the optimization variable in the original optimization problem; by convention, we reFer to x as the primal variables oF the Lagrangian. The second argument oF the Lagrangian is a vector α R m with one variable α i For each oF the m convex inequality constraints in the original optimization problem. The third argument oF the Lagrangian is a vector
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cs229-cvxopt2 - Convex Optimization Overview (cntd) Chuong...

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