gfor the importance sampling density minimizes the variance of and is therefore the solution to the functional minimization program ( ( min varg (xg ( optimal importance sampling pdf it is not difficult to showsee also exercise that if either ( or ( for all xthen the optimal importance sampling pdf is (xh(xf (xu ( namelyin this case varg* varg( (xf ( )/ ( )vargu so that the estimator is constant under gan obvious difficulty is that the evaluation of the optimal importance sampling density gis usually not possiblesince (xin ( depends on the unknown quantity neverthelessone can typically choose good importance sampling density "closeto the minimum variance density gone of the main considerations for choosing good importance sampling pdf is that the estimator ( should have finite variance this is equivalent to the requirement that (xf ( ( ( eg ( (xg(xthis suggests that should not have lighter tails than and thatpreferablythe likelihood ratiof /gshould be bounded monte carlo for optimization in this section we describe several monte carlo methods for optimization such randomized algorithms can be useful for solving optimization problems with many local optima and complicated constraintspossibly involving mix of continuous and discrete variables randomized algorithms are also used to solve noisy optimization problemsin which the objective function is unknown and has to be obtained via monte carlo simulation simulated annealing simulated annealing simulated annealing is monte carlo technique for minimization that emulates the physical state of atoms in metal when the metal is heated up and then slowly cooled down when the cooling is performed very slowlythe atoms settle down to minimum-energy state denoting the state as and the energy of state as ( )the probability distribution of the (randomstates is described by the boltzmann pdf (xf (xekt xwhere is boltzmann' constant and is the temperature