91Ó°ÊÓ

In nonlinear programming problems, the objective function is what you aim to maximize or minimize.
Here, we need to maximize the function given by: \ \(100 - e^{-x} - e^{-y} - e^{-z}\).
Each term involving an exponential function affects the total value.
The higher the values of \(x\), \(y\), and \(z\), the smaller the negative exponential terms become, approaching zero.
This will make the objective function get closer to the value of 100, making it the peak value to aim for.

Constraints limit the feasible region for the variables.
For this problem, the constraints are: \
\(x + y + z \leq a\) \
and
\(x \leq b\).
This means that the sum of \(x\), \(y\), and \(z\) cannot exceed \(a\), and \(x\) itself cannot exceed \(b\). \
The constraints help define the domain where we can search for the maximum value of our objective function.

Partial derivatives measure how a function changes as its input variables change.
For the optimal value function \(f^*(a, b)\), we are interested in its partial derivatives with respect to \(a\) and \(b\).
Computing these gives us: \
\(\frac{\partial f^*}{\partial a}\) and \(\frac{\partial f^*}{\partial b}\). \
These derivatives tell us how sensitive the maximum value of our objective function is to changes in \(a\) and \(b\).
They help to find the inclination or declination rates which are vital for optimization.

Lagrange multipliers are a powerful tool for handling constraints.
They allow us to convert a constrained optimization problem into a form that is easier to analyze.
By introducing a multiplier for each constraint, we adjust the objective function accordingly.\ If \(\lambda\) and \(\mu\) are the multipliers for the constraints \(x + y + z \leq a\) and \(x\leq b\), respectively, the Lagrange function is: \
\(L = 100 - e^{-x} - e^{-y} - e^{-z} + \lambda(a - x - y - z) + \mu(b - x)\).
The relationships between the partial derivatives of \(f^*(a, b)\) and the Lagrange multipliers are given by:\ \(\frac{\partial f^*}{\partial a} = \lambda\) and \(\frac{\partial f^*}{\partial b} = \mu\).

A function is concave if a line segment between any two points on the function lies below or on the curve.
To show that \(F^*(a)\) is concave in \(a\), assume \(b = 0\).
Formally, \(F^*(a)\) is concave if \ \(F^*(\theta a_1 + (1 - \theta) a_2) \geq \theta F^*(a_1) + (1 - \theta) F^*(a_2)\) \ for all \(a_1, a_2\) and \(0 \leq \theta \leq 1\).
As a concave function, increasing \(a\) will always increase the objective function, but at a decreasing rate.