Mathematical Nirvana¶

Problem Definition¶

Nathaniel Richards is a young, brilliant scientist that has developed the following Non-Linear Programming (NLP) problem to find the optimal balance between studying and meditating to maximize the overall satisfaction and achieve a state of enlightenment (Mathematical Nirvana).

Decision Variables: Let:

$x_1$ Time spent studying (in hours)

$x_2$ = Time spent meditating (in hours)

$x = [x_1, x_2]$ The set of decision variables

$x_1, x_2 \geq 0$

Objective Function: Maximize the overall satisfaction obtained from studying and meditating:

$\max z = f(x) = 2*x_1+0.5*\ln(1+x_1) + 0.7*x_2 + 0.3*\sqrt(x_2)$

Constraints: Subject to: Maximum amount of time available:

$x_1 + x_2 \leq 10$

Minimum amount of time studying to ensure academic performance:

$x_1 \geq 2$

Unfortunately Nathaniel mysteriously disappeared before he could completely analyse the problem, so you need to complete his work according to the following instructions:

Obtain the Kuhn-Tucker conditions
Obtain the Hessian and determine if this solution ($x_1 = 3.9, x_2 = 6.1) is a global or local maximum
Use the Kuhn-Tucker conditions to calculate the Lagrangian multipliers for this solution, can you explain what they mean? Discuss if this can be an optimal solution to the problem