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?

A council has a brown-field sites designated for social housing. After allowing for roads and parks it has an area of 150000 m² for social housing. The housing association intends to build a mixture of flats, two-bedroom houses, three-bedroom houses, and shops (which include a living space) on this site. The housing association wishes to maximise the number of people that can be housed but has strict housing requirements: • there must be at least twice as many two-bedroom houses as three-bedroom houses; • at least 10% of the total housing units excluding shops (i.e. flats and houses) must be flats; • buildings containing the flats (i.e a "block") must have two storeys and every flat must have the same floor space; • shops must be provided at a rate of at least one per 1500 inhabitants. The average number of inhabitants per housing unit and the ground floor area of each are given in the following table. Housing type Flat Two-bedroom house Three-bedroom house Shop Average number of…