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convex_opt_art

convex_opt_art - Introduction to Convex Constrained...

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Introduction to Convex Constrained Optimization March 4, 2004 c 2004 Massachusetts Institute of Technology. 1
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1 Overview Nonlinear Optimization Portfolio Optimization An Inventory Reliability Problem Further concepts for nonlinear optimization Convex Sets and Convex Functions Convex Optimization Pattern Classification Some Geometry Problems On the Geometry of Nonlinear Optimization Classification of Nonlinear Optimization Problems Solving Separable Convex Optimization via Linear Optimization Optimality Conditions for Nonlinear Optimization A Few Concluding Remarks 2 Nonlinear versus Linear Optimization Recall the basic linear optimization model: 2
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T LP : minimize x c x s.t. T a 1 x b 1 , · = · . . . T a m x b m , n x . In this model, all constraints are linear equalities or inequalities, and the objective function is a linear function. In contrast, a nonlinear optimization problem can have nonlinear functions in the constraints and/or the objective function: NLP : minimize x f ( x ) s.t. g 1 ( x ) 0 , · = · . . . g m ( x ) 0 , n x , n n In this model, we have f ( x ) : and g i ( x ) : , i = 1 , . . . , m . Below we present several examples of nonlinear optimization models. 3
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3 Portfolio Optimization Portfolio optimization models are used throughout the financial investment management sector. These are nonlinear models that are used to determine the composition of investment portfolios. Investors prefer higher annual rates of return on investing to lower an- nual rates of return. Furthermore, investors prefer lower risk to higher risk. Portfolio optimization seeks to optimally trade off risk and return in invest- ing. We consider n assets, whose annual rates of return R i are random vari- ables, i = 1 , . . . , n . The expected annual return of asset i is µ i , i = 1 , . . . , n , and so if we invest a fraction x i of our investment dollar in asset i , the expected return of the portfolio is: n T µ i x i = µ x i =1 where of course the fractions x i must satisfy: n T x i = e x = 1 . 0 i =1 and x 0 . (Here, e is the vector of ones, e = (1 , 1 , . . . , 1) T . The covariance of the rates of return of assets i and j is given as Q ij = COV( i, j ) . We can think of the Q ij values as forming a matrix Q , whereby the variance of portfolio is then: n n n n COV( i, j ) x i x j = Q ij x i x j = x T Qx. i =1 j =1 i =1 j =1 It should be noted, by the way, that the matrix Q will always by SPD (symmetric positive-definite). 4
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The “risk” in the portfolio is the standard deviation of the portfolio: STDEV = x T Qx , and the “return” of the portfolio is the expected annual rate of return of the portfolio: T RETURN = µ x . Suppose that we would like to determine the fractional investment values x 1 , . . . , x n in order to maximize the return of the portfolio, subject to meet- ing some pre-specified target risk level. For example, we might want to ensure that the standard deviation of the portfolio is at most 13 . 0%. We can formulate the following nonlinear optimization model: T MAXIMIZE: RETURN = µ x s.t.
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