one of the test cases in [ involves the minimization of the hougen function implement cross-entropy and simulated annealing algorithm to carry out this optimization task in the binary knapsack problemthe goal is to solve the optimization problemmax pxx{ , } subject to the constraints ax cwhere and are vectors of non-negative numbersa (ai is an matrixand is an vector the interpretation is that or depending on whether item with value is packed into the knapsack or not nthe variable ai represents the -th attribute ( volumeweightof the -th item associated with each attribute is maximal capacitye could be the maximum volume of the knapsackc the maximum weightetc write ce program to solve the sento dat knapsack problem at le brunel ac uk/~mastjjb/jeb/orlib/files/mknap txtas described in [ let ( )( )be renewal reward processwith er and pnt ec let at = ri / be the average reward at time where nt max{ tand we have defined ni= ci as the time of the -th renewal (ashow that / -ec as (bshow that nt -as (cshow that nt / - /ec as [hintuse the fact that nt nt + for all (dshow that at -er as ec prove theorem prove that if ( the importance sampling pdf gin ( gives the zerovariance importance sampling estimator let and be random variables (not necessarily independentand suppose we wish to estimate the expected difference [ yex ey (ashow that if and are positively correlatedthe variance of is smaller than if and are independent (bsuppose now that and have cdfs and grespectivelyand are simulated via the inverse-transform methodx - ( ) - ( )with uv ( )not necessarily independent intuitivelyone might expect that