sum of absolute values constraint in semi definite programming

Question

I want to further my real world semi definite programming optimization problem with a constraint on sum of absolute values. For example:

abs(x1) + abs(x2) + abs(         


        

Answer 1

I have also seen the variation:

x_1 = x1_plus - x1_minus

x2 = x2_plus - x2_minus xN = xN_plus - xN_minus and x1_plus, x1_minus>=0 x2_plus, x2_minus>=0

xN_plus, xN_minus>=0

and

x1_plus+x1_minus + x2_plus+x2_minus + ... + xN_plus+xN_minus <=10

Cost: an additional 2N variables, N equality constraints + 2N+1 inequality constraints. Much more than @fanfan's but with other benefits.

xk_plus and x_k_minus can be used in an objective function to penalize either the absolute value of xk or to give different penalties for the positive and negative parts of xk. These slack variables are sometimes used to express transaction costs, e.g.,

max theta'mu - lambda/2 theta'Sigma theta -TC(buy+sell)

theta = theta0+buy-sell

buy,sell>=0

and allow for an asymmetry in TC if required.