0
votes
0answers
14 views

Eliminating variables in convex program

This is a basic convex optimisation question. I have the following problem: $$\max_{\substack{t\le e\\ At\le b}} e^\top t$$ How do I find the optimum $t^*$? I write the KKT conditions, get ...
0
votes
0answers
16 views

Let $P$ be a polyhedron. Prove, $P$ has at least one extreme point $\iff$ $P$ does not contain a line, by using a lemma.

Let $P$ be a polyhedron. Prove, $P$ has at least one extreme point $\iff$ $P$ does not contain a line, by using a lemma. I've a Lemma saying: Suppose $P=P(V,E)$ where $V,E \in \mathbb R^n$ are ...
0
votes
0answers
13 views

Let $P \subseteq R^n$ be a polyhedron. Why does $\{ x + \alpha d \mid \alpha > 0\} \subseteq P$ for some $x \in P$ imply $d$ is a recession direction?

Suppose we have a polyhedron $P \subseteq R^n$ and let $d \in P$ be a recession direction, that is $\{ x + \alpha d \mid \alpha > 0\} \subseteq P$ for all $x \in P$. Why does $\{ x + \alpha d \mid ...
0
votes
1answer
23 views

Write the convex hull of the points $(0,0), (1,1), (2,0) \in \mathbb R^2$ as the set of solutions to a finite number of inequalities.

Write the convex hull of the points $(0,0), (1,1), (2,0) \in \mathbb R^2$ as the set of solutions to a finite number of inequalities. The convex hull of $x_1, \ldots, x_n \in \mathbb R^n$ is defined ...
0
votes
1answer
32 views

what is the dual of the following linear program over a convex set?

Let $\mathbf{x}=[x_0,x_1,\dots,x_N]^T$ be a $(N+1)\times 1$vector. Let $\mathcal{S}$ be a bounded, compact convex set in strictly positive quadrant of $\mathbb{R}^{N+1}$. Consider the following ...
2
votes
1answer
54 views

Prove vertices of a simplex are affinely independent

I'm given that the definition of a simplex $T$ is $x \in\mathbb R^n$ such that $x$ satisfies $n+1$ linear inequalities: $(u_k, x) \lt c_k$ for $k = 1,\ldots,n+1$ (i.e. $T$ is the intersection of $n+1$ ...
0
votes
1answer
39 views

Projection onto Polyeder

I know how to projects onto a linear subspace of $\mathbb R^3$, but how to project a point $x$ onto an polyhedron given as the intersection of three halfspaces $$ \langle y_1, x \rangle \ge c_1 ...
0
votes
0answers
33 views

Union of all sets of optimal solutions to a perturbed linear programming problem

Please let me know if you have some ideas on how to approach this proof? I got stuck part-way through. The following linear program is a function of $\theta$, $ \begin{array}{ll} \min & c^\top x ...
0
votes
0answers
20 views

Robust LP question using box uncertainty model

I am trying to solve this robust LP problem by writing it as a QP $$\min_x x^TSx : \mu \leq r^T x , Ax \leq b$$ Under Box uncertainty model: $$R = \{r : \| r - \hat{r}\|_\infty \leq \rho\}$$ Here ...
1
vote
2answers
60 views

Minimization of log-sum-exponential function subject to constraints.

I would like to minimize the following function: $f(x)=log(e^{-x_1}+..+e^{-x_n})$ Subject to: $\sum_{i=1}^{n}{x_i}=1$ $0 \leq x_i \leq 1$ So far I have discovered the following: If all the ...
0
votes
0answers
52 views

Online convex programming: Projection followed by normalization

I have the following projected gradient descent online linear programming problem which has been well studied in www.cs.cmu.edu/~maz/publications/techconvex.pdf‎ $\mathbf{y}_{t+1}=\mathbf{w}_t - ...
2
votes
0answers
50 views

convexity in oriented matroid theory

I would like to try to solve the following problems. If someone knows how to prove at least part (a), could you show me the proof? I am having a LOT of trouble understanding oriented matroid theory. ...
0
votes
1answer
193 views

Finite optimal value for a linear program with unbounded feasible region.

I read this problem in CLRST : Show that a linear program can have finite optimal objective value even if the feasible region is not bounded. Now all the cases I could think of where such a thing ...
0
votes
0answers
96 views

Proof of corollary of Farkas' lemma

I tried to prove the following lemma of Farkas' lemma: Given the system $Ax<b$, $A\in \mathbb{R}^{m\times n}$, $b\in \mathbb{R}^m$, the system is infeasible iff there exists $\lambda\in ...
1
vote
2answers
20 views

Linear program dual

We are trying to find the dual of the following linear program. $$ \max_x \ 2x_1 \ + x_2 \ \ \ \ -- (1) $$ such that, $$ x_1 + x_2 \leq 2 \ \ \ \ -- (2)\\ -x_1 - x_2 \leq -4 \ \ \ ...
1
vote
1answer
117 views

Determining the convex hull of the union of two polyhedra

I'm doing an introductory course to linear programming and I'm working through some exercises to prepare for the final exam, I'm stuck on an exercise and I would really appreciate a hint: Let ...
0
votes
1answer
30 views

Find a point in a polytope that always cuts off a constant fraction of the polytope.

I have $k$ linear inequalities in $\mathbb{R}^n$, which we can express as $Ax \ge c$ (where $A \in \mathbb{R}^{k \times n}$ and $c \in \mathbb{R}^k$). Assume that the set of $x \in \mathbb{R}^n$ that ...
2
votes
1answer
170 views

Convex optimization and linear programming please help! :)

How would I write the following as a standard form LP? Minimizing $\sum_{i=1}^n x_i + c\max(a_i-x_i)$ for $a_i \ge 0$ and what is the optimal value for when $c=n$ How to express minimize $\frac{1}{2} ...
4
votes
0answers
52 views

Mappings preserving convex polyhedra

It is known that linear mappings between euclidean spaces map convex polyhedra to convex polyhedra. Can you give a characterization of the class of mappings that preserve convex polyhedra?
2
votes
1answer
49 views

Is this a linear programming problem

If $x \in R^n$, then $\min \|x\|_{\infty}$ sub to $Ax = b$, $x \geq 0$ where $\|x\|_{\infty}$ is the infinity norm which is $\max\{\|x_1\|,\|x_2\|,\ldots,\|x_n\|\}$. If not then how can ...
3
votes
1answer
93 views

A particular ILP where the existence of a relaxed solution implies the existence of an integer solution

This question emerged from a discussion on my previous question Determining quickly whether this Integer Linear Program has any solution at all, which I would like to elaborate separately. I am ...
4
votes
5answers
500 views

Find a convex combination of scalars given a point within them.

I've been banging my head on this one all day! I'm going to do my best to explain the problem, but bear with me. Given a set of numbers $S = \{X_1, X_2, \dots, X_n\}$ and a scalar $T$, where it is ...
0
votes
2answers
273 views

Convex function from Hessian

Am I correct to say that the following function is convex? $$\begin{align} & f(x,y)=-\sqrt{xy} \\ & x>0,y>0 \\ \end{align}$$ After computing the Hessian: $$ Hf =\left[ \begin ...
6
votes
0answers
740 views

Farkas Lemma proof

I am trying to prove the Farkas Lemma using the Fourier-Motzkin elimination algorithm. From Wikipedia: Let A be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the ...
1
vote
1answer
394 views

Convex hull of sets defined by (in)equalities

If you define the convex hull of a set $X$ as the set of all convex combinations of elements of $X$, it becomes difficult to decide if a given element $w$ belongs or not to $conv(X)$ (You have to ...
4
votes
1answer
655 views

Proving Helly's theorem

The problem is to prove Helly's theorem for the case, when the convex bodies are polytopes, by using linear programming duality. Helly's theorem Let $B_{1},...,B_{m}$ be a collection of convex ...