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Problem 4:Piecewise linear functions(18 marks)(a) Consider the followingclustering problem. We are givenkpointsp(1), p(2), . . . , p(k)∈Rn. The goal isto find a “center” pointx∈Rnthat best “summarizes” the data of thekpoints. To model this, we definethe dissimilarity between two pointsa, b∈Rnas follows:dis(a, b) :=∑ni=1|ai-bi|. For example, ifa= (1,2.4,0,-2.3)Tandb= (2,-1,1.2,0)Tare points inR4, thendis(a, b) =|1-2|+|2.4-(-1)|+|0-1.2|+| -2.3-0|= 7.9. (Note thatdis(a, b) = dis(b, a).)Our goal now is to find a pointx∈Rnthat minimizesmaxj=1,...,kdis(x, p(j)); that is, we want to findx∈Rnthat minimizesmaxj=1,...,k∑ni=1|xi-p(j)i|. Formulate this as a linear program.(10 marks)8
(b) For each of the following, indicate whether the given mathematical program and the proposed LP formu-lation are equivalent, that is, whether they have the same optimal values. Give a reason for your answer.(8 = 4×2 marks)Mathematical programLP Formulation