Tagged Questions
2
votes
1answer
28 views
Formulate the Langrangian function of a non-linear optimization problem and solve it for $y\geq0$
Consider the (non-linear) optimization problem ($P$)
$$max \quad3x_1 + 4x_2$$
$$s.t. \quad x_1^2 + x_2^2 \leq 25$$
$$ \quad x_1,x_2 \geq 0$$
Formulate the Lagrangian function ...
1
vote
1answer
67 views
Cost minimization problem
The problem is as follows:
A firm uses $k$ units of capital and $l$ units of labor to produce
$(k^{\alpha}l^{1-\alpha})^{1/\beta}$
units of output, where $\alpha$ and $\beta$ satisfying $0 < ...
1
vote
0answers
41 views
Formulation of a problem as semidefinite programming
I would appreciate some help with this problem:
$R$ is a positive semidefinite matrix $\in{R}^{n\times n}$, $A \in{R}^{n\times m}$.
I need to formulate this optimization problem as semidefinite ...
4
votes
1answer
109 views
Kuhn-Tucker condition is not satisfied
Show that the solution to finding minimum of
$f(x)=-x_{1}$
With conditions
$-\sin(x_{1})+x_{2} \leq 0$
$x_{1}-x_{2} \leq 0$
is point $(0,0)$, but the Kuhn-Tucker condition is not satisfied in this ...
0
votes
0answers
29 views
checking whether or not a function is concave
Given $t$ as a random variable with a smooth distribution $f(t)$ which is independent of $y_{i}$ and a function
$f(x)=g(y₁)-αE[max(y₁,y₂)]-βE[max(y₁+t-x,0)]$
with the following properties:
...
1
vote
1answer
133 views
A sufficient condition for a unique maximum of the product of two concave functions
Given two concave functions $f(x)$ and $g(x)$, what conditions in terms of these functions can ensure that
$h(x)=f(x)g(x)$
have a unique maximizer on an interval $[a,b]$ for $a<b$?
1
vote
0answers
99 views
sufficient condition for KKT problems
For the Karush-Kuhn Tucker optimsation problem, Wikipedia notes that:
"The necessary conditions are sufficient for optimality if the objective function f and the inequality constraints g_j are ...
0
votes
0answers
69 views
Unique utility, but variable Walrasian Equilibrium Consumption set
Basically, I am not sure why the utilities in this economy is unique even though Walrasian Equilibrium consumption is not unique.
Here's the setup:
Consider an exchange economy with one money good ...
0
votes
0answers
62 views
Iterative scheme for a nonlinear optimization problem
Let $\mathbb{PD}_3 \subset \mathbb{R}^{3 \times 3}$ be the set of the positive-definite $3 \times 3$ real matrices.
For given $v \in \mathbb{R}^{3 \times 1}$, consider the function $f_v: ...
0
votes
0answers
40 views
A question in non linear optimization.
If we consider the linearly constrained optimization problem
min f(x), A^T.x < b. A is an n X m matrix.
The set of feasible solutions is S and for x in S, I(x) is the set of active constraints.
...
1
vote
1answer
87 views
Question regarding Kuhn-Tucker multiplier
I have a problem which I am unable to solve. If we consider the following problem
$\min f(x)$,
$G(x) = b$;
where $f$ is in $C^2(R^n)$, and $G$ from $R^n$ to $R^m$ is a $C^2$-function, $G = ...
0
votes
1answer
322 views
Set of symmetric positive semidefinite matrices is a full dimensional convex cone.
If $S_n^+$ is the set of all symmetric positive semidefinite $n \times n$ matrices with entries in $\mathbb{R}$, how does it follow that it is a full dimensional closed convex cone in ...
