FE501 – 1221 NLP Inequality and Equality Case

Information about 2nd mid-term

1. Portfolio optimiation problems are also constrained NLPs.

2. In the mid-term there will be 4 questions:

   1. Solve an unconstrained Nonlinear problem (1214)
   2. Solve an constrained NLP with equality constraints (1221)
   3. Solve an constrained NLP with inequality constraints (1221)
   4. Showing convexity or concavity of the functions (1207)

Constrained NLPs with Equality Constraints

General form of a constrained NLPs with equality constraints is:

Suppose we have such equality constraint NLP as below:

Solve (1) with graphical solution precedure

1. draw the constraint of the problem

   

2. Draw isoprofit (isocost) line (curve) correspondingly

   (How can we draw all the points which gives us the objective values of 0 ?)

   We take the gradient of the $\mathbf{\nabla }f$$\grad f$

   $$\begin{array}{}\text{(2)}& \begin{array}{c}\mathbf{\nabla }=\left[\begin{array}{c}\frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial y}\end{array}\right]=\left[\begin{array}{c}1\\ 1\end{array}\right]\end{array}\end{array}$$

   the gradient vector is always perpendicular to the level curve.

From the graph we can find that, the $z=x+y$$z=x+y$ takes maximum value (optimal value ) at the intersrct point (c,x+y) at A(1,1)

At the local optimal point, the gradient of the objective function and the gradient of the constraint function is parallel.

Gradient of the objective function:

Gradient of the constraint :

Lagrangian Multiplier

Lagrange multipliers can be used to solve NLPs in which all the constraints are equalityconstraints. We consider NLPs of the following type:

To solve (eq), we associate a multiplier  ${\lambda }_i$$\lambda _i$ with the $i$$i$th constraint in (eq) and form the Lagrangian

Notice: The number of $\lambda$$\lambda$ is same as the number of equality functions (that is to say $m$$m$)

Then we attempt to find a point $(x_1,x_2,...,x_n,{\lambda }_1,{\lambda }_2,...,{\lambda }_m)$$(x_1, x_2, . . . ,x _n,\lambda_1,\lambda_2, . . . ,\lambda_m)$ that maximizes (or minimizes) $L(x_1,x_2,...,x_n,{\lambda }_1,{\lambda }_2,...,{\lambda }_m)$$L(x_1,x_2, . . . , x_n,\lambda_1,\lambda_2, . . . ,\lambda_m)$.

That is to say, when we will solve

Solve (1) with Lagrangian Multiplier

Lagrangian Multiplier in Advertising

A company is planning to spend $10,000 on advertising. It costs $3,000 per minute to advertiseon television and $1,000 per minute to advertise on radio. If the firm buys $x$$x$ minutesof television advertising and $y$$y$ minutes of radio advertising, then its revenue in thousandsof dollars is given by $f(x,y)=-2x^2-y2+xy+8x+3y$$f (x, y)=-2x^2-y2+xy+8x+3y$. How can the firmmaximize its revenue?

Solve (7) without the constraint

Solve (7) with equality constraint

Unconstrained inequality NLPs (KTT)

General form

To apply the results of this section, all the NLP’s constraints must be$\le$$\le$constraints.

Other types of constraines must be changed, for example:

Lagrangian Function of inequality

For minimization problem:

For maximization problem:

KKT Conditions

1. Stationary conditions

   partial derivatives set to zero

2. primal feasibility

   original constraints must satisfy

3. complementary slackness

   dual must be zero :

   1. strict ? $\mu =0$$\mu =0$
   2. not strict ? $b_i-g_i=0$$b_i - g_i = 0$

4. dual feasibility

   $\mu$$\mu$ is not negative.

Expl.1. max $x_1-x_2$$x_1 - x_2$

Expl.2. min $x_1-x_2$$x_1 - x_2$

Expl.3. 50 points problem (😅)

Note: I do have calculated the problem, but, i have a completly different answer to that. To not mislead anyone, i put on the original answers, to check my answers, you may click here.

Max x*y

Solving using the graphical precedure

Using KKT

End of 1221.