Consider the following two minmax optimization problems in two variables inf sup V(x, y) XeX YEY...

 

Transcribed Image Text:

Consider the following two minmax optimization problems in two variables inf sup V(x, y) XeX YEY sup inf V(x, y) yeYxex in which x = X = R", yeYER", and V: XXY - R. (a) Show that the values are ordered as follows inf sup V(x, y) sup inf V(x, y) XeX YEY or, if the solutions to the problems exist, yeYXEX min max V(x, y) max min V(x, y) yeY xex A handy mnemonic for this result is that the player who goes first (inner problem) has the advantage.9 "Note that different conventions are in use. Boyd and Vandenberghe (2004, p. 240) say that the player who "goes" second has the advantage, meaning that the inner prob- lem is optimized after the outer problem has selected a value for its variable. We say that since the inner optimization is solved first, this player "goes" first. 768 (b) Use your results to order these three problems sup inf sup V(x,y,z) XeX YEY ZEZ inf sup sup V(x, y, z) yey zez xex Optimization sup sup inf V(x,y,z) zel xex yey Consider the following two minmax optimization problems in two variables inf sup V(x, y) XeX YEY sup inf V(x, y) yeYxex in which x = X = R", yeYER", and V: XXY - R. (a) Show that the values are ordered as follows inf sup V(x, y) sup inf V(x, y) XeX YEY or, if the solutions to the problems exist, yeYXEX min max V(x, y) max min V(x, y) yeY xex A handy mnemonic for this result is that the player who goes first (inner problem) has the advantage.9 "Note that different conventions are in use. Boyd and Vandenberghe (2004, p. 240) say that the player who "goes" second has the advantage, meaning that the inner prob- lem is optimized after the outer problem has selected a value for its variable. We say that since the inner optimization is solved first, this player "goes" first. 768 (b) Use your results to order these three problems sup inf sup V(x,y,z) XeX YEY ZEZ inf sup sup V(x, y, z) yey zez xex Optimization sup sup inf V(x,y,z) zel xex yey