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Chapter 7: Q9E (page 240)

A quadratic programming problem seeks to maximize a quadratic objective function (with terms like 3x12or5x1x2) subject to a set of linear constraints. Give an example of a quadratic program in two variables x1, x2 such that the feasible region is nonempty and bounded, and yet none of the vertices of this region optimize the (quadratic) objective.

Short Answer

Expert verified

The required better fit example is shown below:

max x1x2 subject to

x1+x2鈮�1x1,x2鈮�0.

Step by step solution

01

Consider an example

Consider the maximum condition x1x2.

Subject to, x1+x2鈮�1x1,x2鈮�0.

02

Solve the quadratic programming

It is known that, maxx1x2 subject to

x1+x2鈮�1x1,x2鈮�0.

The constraints can be written as,

g1=-x1-x2+1鈮�0g2=x1鈮�0g3=x2鈮�0

It can be easily observed that the maximum is obtained at the point (x1,x2)=12,12, which implies that,

x1x2=12.12=14

Where the vertices of the feasible region are (0,0),(1,0),(0,1).

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Most popular questions from this chapter

Hollywood. A film producer is seeking actors and investors for his new movie. There are n available actors; actori chargesSj dollars. For funding, there arem available investors. Investorj will providepj dollars, but only on the condition that certain actorsLj鈯�{1,2,...,n} are included in the cast (all of these actorsLj must be chosen in order to receive funding from investorrole="math" localid="1658404523817" j ).

The producer鈥檚 profit is the sum of the payments from investors minus the payments to actors. The goal is to maximize this profit.

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The pizza business in Little Town is split between two rivals, Tony and Joey. They are each investigating strategies to steal business away from the other. Joey is considering either lowering prices or cutting bigger slices. Tony is looking into starting up a line of gourmet pizzas, or offering outdoor seating, or giving free sodas at lunchtime. The effects of these various strategies are summarized in the following payoff matrix (entries are dozens of pizzas, Joey鈥檚 gain and Tony鈥檚 loss).




TONY




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What is the value of this game, and what are the optimal strategies for Tony and Joey?

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