This is a theorem, please prove it Show that the KKT conditions when applied to LP give the certificate of optimality "Primal feasibility, dual feasibility, and complementary slackness."

A flood control reservoir project and channel improvement project are proposed for reducing flood damages along a stream in an urban area. Estimated costs are as follows: Dam and reservoir project Channel improvement project Initial investment $1,500,000 $1,200,000 Annual maintenance $10,000/yr $5,000/yr Hydrologic, hydraulic and economic studies have been conducted to develop the following relationships for the river reach and flood plain: (a) annual exceedance probability versus peak discharge with and without the reservoir, (b) discharge versus stage with and without the channel improvement, and (c) stage versus damage Annual Peak discharge Stage (ft) exceedance (cfs) probability Without With Discharge reservoir reservoir (cfs) Without With Stage Damage project project (ft) million) 0.50 3,000 2,700 0 0 0 9 0.0 0.20 5,400 0.10 7,600 4,200 3,500 5,000 10. 7 10 0.2 10,000 15° 11 15 1.1 0.05 10,400 5,500 15,000 20 14 20 2.2 0.02 0.01 19,000 0.005 24,000 20,000 15,000 8,600 20,000 23 17…

Your town is considering updating one of the local parks. They created a linear program that had the following constraints: 1. Spend no more than $100,000 2. Spend no more than $50,000 on new playground equipment 3. Put in at least three basketball courts After running the linear program, the town saw that the program was infeasible. The town has decided to make constraints 1 & 2 goals, instead of hard constraints. How many decision variables would need to be added to the town's linear program, in order to turn it into a goal program with two soft constraints that allow deviation in both directions?

Calculating outcomes as equally likely would BEST describe: O a. Maximax criterion O b. Laplace criterion O c. Regret criterion Od. Maximin criterion Determining the average payoff for each alternative and choosing the one with the BEST payoff is the approach called: ea, maximax O b. minimax regret O c. laplace Od maximin M