0
votes
1answer
20 views

Maximization with the Dual using the Simplex Method.

I have an exam in a few hours. I need to understand the solution to the following question Find the Maximal to the the following $2 x_1 + 3 x_2$ is the objective function. The constraints are ...
0
votes
1answer
44 views

Lagrangian dual for the sum of norms

I would like some help in deriving the Lagrangian dual function of a sum-of-norms minimization problem : $\sum{||A_{i}x-b_{i}||}$ when $A_{i}$ are matrices, and $b_{i},x$ vectors. I understand I can ...
0
votes
0answers
21 views

how to check slater condition for a constrained optimization problem?

Given any optimization problem that you suppose to solve with Lagrange by thrusting strong duality, you need to be sure the Slater Conditions. And I guess there is no algorithmic way to solve for all ...
1
vote
0answers
49 views

Duality in Chebychev approximation

I got messed up with this problem and can't find any clue to solve this. Hope some one here can help me. Let $A$ be an $m \times n$ matrix an let $b$ be a vector in $R^{m}$. We consider the ...
6
votes
1answer
146 views

Is duality theory in optimization as useful as it seems?

I have been reading a lot about nonlinear optimization and duality, and it seems that duality theory is extremely useful. I feel that I am missing some of the negative aspects/difficulties associated ...
1
vote
3answers
145 views

What is the dual of this optimization problem?

Consider the points $x_1, \ldots, x_N \in \mathbb{R}^n$, and a (locally bounded, convex) function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. I am looking for the dual of the following optimization ...
1
vote
1answer
150 views

Are these convex optimization problems equivalent?

Consider the optimization problem $$ \mathcal{P}_1: \qquad \min_{x \in \mathbb{R}^n} c^\top x \quad \text{sub. to } \ g(x,y_i) \leq 0 \ \ \forall i = 1,2,...,M$$ where $c \in \mathbb{R}^n$, and ...
0
votes
1answer
228 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
2
votes
3answers
113 views

Dual of a Linear Program

\begin{align} \min_{x} c^Tx \\ s.t.~Ax=b \end{align} Note that here $x$ is unrestricted. I need to prove that the dual of this program is given by \begin{align} \max_{\lambda} \lambda^Tb \\ ...
0
votes
1answer
556 views

Maximizing and Minimizing a function

Let $f(x,y)$ be a function such that $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$. Now we have to maximize $f$ over $x$ and minimize it over $y$ $i.e.\ $ $$\underset{x}{\text{max}}\: ...
0
votes
1answer
238 views

Underlying assumption in a Primal/Dual table

I just read in one of the questions answered by @MikeSpivey that the following table is provided in Sierksma's Linear and Integer Programming: Theory and Practice, Volume 1, page 144. ...
1
vote
1answer
159 views

Multiple solutions for both primal and dual

If matrix $A$ in an LP (or $A^T$ in its dual) has full row (column- in dual) rank, is it possible that both primal and dual have multiple solutions?
6
votes
1answer
173 views

Difficulties in Writing the Dual of a Primal Program

I am a student and I am studying the following problem during my spare time. Your comments and suggestions would be helpful. Given the following primal program: (Decision variables are $\xi_{v}$, ...
5
votes
1answer
429 views

Conditions for no duality gap in quadratic programming?

Assume $Q \in \mathbb{R}^{n\times n}$, and $b,c,d \in \mathbb{R}^n$. A quadratic programming problem is: $$ \min_{x \in \mathbb{R}^{n}} \tfrac{1}{2} x^T Q x + c^T x,$$ subject to $A x \leq b, E x ...
3
votes
0answers
71 views

Questions about no duality gap and optimal solutions

My questions are regarding a constrained problem, $$\min_{x \in X \subseteq \mathbb{R}^n} f(x),$$ subject to $g(x) \leq 0 \in \mathbb{R}^m, h(x) =0 \in \mathbb{R}^k$. Its dual problem is $$\sup_{u ...
1
vote
1answer
216 views

Questions about weak duality theorem

Following are some corollaries regarding the weak duality theorem. Consider a constrained problem, $\min_{x \in X} f(x),$ subject to $g(x) \leq 0$ and $h(x) =0$. Its dual problem is $\sup_{u \geq ...