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R Solution. R - Stanford Engineering Everywhere

EE364a, Winter 2007-08 Prof. S. BoydEE364a Homework 4 involving 1- and -norms. Formulate the following problems as LPs. Ex-plain in detail the relation between the optimal solution of each problem and thesolution of its equivalent LP.(a) MinimizekAx bk ( -norm approximation).(b) MinimizekAx bk1( 1-norm approximation).(c) MinimizekAx bk1subject tokxk 1.(d) Minimizekxk1subject tokAx bk 1.(e) MinimizekAx bk1+kxk .In each problem,A Rm nandb Rmare given. (See for more problemsinvolving approximation and constrained approximation.)Solution.(a) Equivalent to the LPminimizetsubject toAx b t1Ax b the variablesx Rn,t R. To see the equivalence, assumexis fixed in thisproblem, and we optimize only overt. The constraints say that t aTkx bk tfor eachk, ,t |aTkx bk|, ,t maxk|aTkx bk|=kAx bk.

We consider a linear dynamical system with state x(t) ∈ Rn, t = 0,...,N, and actuator or input signal u(t) ∈ R, for t = 0, ... for actuator signals between −1 and 1; for larger actuator signals the marginal fuel efficiency is half. Formulate the minimum fuel optimal control problem as an LP.

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