Question

X is an n x 1 state vector with n decision variables. The n decision variables relate to quantities of electronic components produced by the manufacturer based at Sensor City. The profit the manufacturer expects from producing these components is given by the following quadratic function, f(x) = {x"QX+Bºx + c where Q is a negative definite matrix and B and C are constant vectors. Find the state vector, X, that maximises the profit, f(x), when there are no constraints placed on the quantities produced. ii) Due to external factors relating to a supplier based at Wavertree Technology Park, the quantity of components that can be produced is constrained by, SX = p where is an m x n matrix, m <n, and p is a constant vector. i) Find the state vector, X, that maximises the profit, f(x), given this constraint.

Answer 1

1 Soli Given > is an nx1 state vector with n decision variables. These components given by the following quadratic function. f(3) = {xtQx+85x + c where, B and care constant vector a is negative definate matrix. i) Given State vector x, that maximises the profit, we find the f(x). > Given problem is maximization problem converting if into maximization problem F* (*) = -f(x) f*(x)=-XTQx+B x + - when no constraints are placed on quantities produced. → necessary condizion we have, df(T) dx ::df*(8) dx - (ax +zła)-B":o

2 NOW, Q is negative definite matrix Que tała = 2xQ3 -Q2-8"=0 multiplying a on both sides. X = -Q B It will give minimum profile of f*Q) and maximum FGC) ii) Given, → The quantity of components that can be produced is constrained by. 1SX =P = BX-P=0 NOW, we using language method La f* () +11(x) L=-(xTQX+B'x+c)+1(sx-P) whenie LCO) - equality constraint leb we find minimum value of y' >> necessary condition will be dL da =O dL di

3 dL dx = '{ (@x+x?@)-8"+d5=0 Q' is negative definite. Qx+xTQ = 2Qx idL dx = Qx-8 + 19 =0 Now, dL dd =SX-P 20 &x=5'p Now, substihefing above eqn cQs'p-BT+dS=0 11- (s'e+B") 5' *. It will give minimum for f(x) and maximum for f(DL). Hence proved