2. Robust optimization You are the owner of a factory that makes widgets and gadgets. For each widget and gadget, there is a set of feature measurements, e.g. the weight, dimensions, material composition, etc., packed in a vector x. For maximum profit, you want the combinations of these features to exactly match a set of gold standard specifications, e.g. Ax -b where A dictates how the features are combined, and b is the gold standard. However, realistically, it is too expensive to exactly satisfy Ax -b, so you have to make some concessions. Instead, you can solve one of three optimization problems minimize ||Ax - b|li- minimize |A-bl max ak-bk mininize A- (a) Show that (1) is a linear program by writing it in standard LP form (b) Show that (2) is a linear program by writing it in standard LP form (c) In Julia, fill in the blanks to solve the LPs. Compare the histogram of the residuals e- Ar -b where x* is the solution for each of the three problems. What conclusions can you draw about the effect of each norm? (d) Widgets are used in the health industry, and if Iakw-bk! is too large, then patients with illness k will die. You must ensure that not a single patient meets this terrible fate. Which norm do you pick? (e) Gadgets are used in the retail industry, and afz - bk means customers of market segment k is satisfied. Realistically, not all customers can be satisfied, but we want to maximize as many market segments as possible. Which norm do you pick? 2. Robust optimization You are the owner of a factory that makes widgets and gadgets. For each widget and gadget, there is a set of feature measurements, e.g. the weight, dimensions, material composition, etc., packed in a vector x. For maximum profit, you want the combinations of these features to exactly match a set of gold standard specifications, e.g. Ax -b where A dictates how the features are combined, and b is the gold standard. However, realistically, it is too expensive to exactly satisfy Ax -b, so you have to make some concessions. Instead, you can solve one of three optimization problems minimize ||Ax - b|li- minimize |A-bl max ak-bk mininize A- (a) Show that (1) is a linear program by writing it in standard LP form (b) Show that (2) is a linear program by writing it in standard LP form (c) In Julia, fill in the blanks to solve the LPs. Compare the histogram of the residuals e- Ar -b where x* is the solution for each of the three problems. What conclusions can you draw about the effect of each norm? (d) Widgets are used in the health industry, and if Iakw-bk! is too large, then patients with illness k will die. You must ensure that not a single patient meets this terrible fate. Which norm do you pick? (e) Gadgets are used in the retail industry, and afz - bk means customers of market segment k is satisfied. Realistically, not all customers can be satisfied, but we want to maximize as many market segments as possible. Which norm do you pick