additional_exercises_sol-homework_17-18.pptx - with the...

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with the natural domain ( i.e. , defined by g ( x ) < o ) . In one simple interpretation, f i ( x i ) is the cost for the i th firm to produce a mix of products given by x i ; g ( x ) is then the optimal cost obtained if the firms can freely exchange products to produce, all together, the mix given by x . (The name ‘convolution’ presumably comes from the observation that if we replace the sum above with the product, and the infimum above with integration, then we obtain the normal convolution.) (a) Show that g is convex. (b) Show that g * = f 1 * + . . . + f m * . In other words, the conjugate of the infimal convolution is the sum of the conjugates. Solution. (a) We can express g as x 1 ,...,x 1 7 g ( x ) = inf ( f 1 ( x 1 ) + . . . + f m ( x m ) + φ ( x 1 , . . . , x m , x )) , where φ ( x 1 , . . . , x m , x ) is 0 when x 1 + . . . + x m = x , and o otherwise. The function on the righthand side above is convex in x 1 , . . . , x m , x , so by the partial minimization rule, so is g . (b) We have x T g * ( y ) = sup( y x - f ( x )) x T = sup y x - x 1 + ... + x = x inf f 1 ( x 1 ) + . . . + f m ( x m ) = sup y T x 1 - f 1 ( x 1 ) + . . . + y T x m - f m ( x m ) , x = x 1 + ... + x where we use the fast that ( - inf S