All Questions
480 questions with no upvoted or accepted answers
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322
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Decomposition of Polyhedral - An example
There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system
$$ \...
- 111
1
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1
answer
173
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Taut string algorithm and TV-minimization equivalence
Given real numbers $y_i's$, consider the following convex optimization problem:
$$
\min_{x_i's} \sum_{i=1}^N(y_i-x_i)^2 + \lambda\sum_{i=1}^{N-1}|x_{i+1}-x_{i}|.
$$
The paper A Direct Algorithm for 1D ...
- 1,216
1
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0
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113
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Statistical analysis of optimization solution involving Brenier potentials?
I'm reading the paper https://arxiv.org/pdf/1905.10812.pdf where strongly convex approximations to Brenier potentials are approximated.
Let $\mathcal{E}$ be a partition of $\mathbb{R}^{d}$ and $ 0\leq ...
- 383
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47
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Prove that a polygon is convex over a circle
The problem
Let $C_A$ (resp. $C_B$) a circle of center $A = (x_A,0)$ (resp. $B = (x_B,0)$) and radius $r_A$ (resp. $r_B$).
For $k = 0,1,2,3,4$, let $D_k$ some points on $C_A$ with $D_0 = (x_0,0)$
Let $...
- 131
1
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52
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Minimum operator that exceeds others (in a PSD, linear matrix inequality, sense)
Given a collection of $n$ matrices $A_i$, we could ask for the $B$ such that:
$$\textrm{Minimize: }\quad \textrm{Tr}[B]$$
$$\textrm{Such that: }\forall_i\, B \succeq A_i$$
Here $\succeq$ is in the ...
- 1,203
1
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0
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48
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Various definitions of coercivity
In this post one says that a functional $F:H\rightarrow [0,\infty]$ on an infinite-dimensional Hilbert space $H$ is (strongly) coercive if there exists a constant $k>0$ such that
$$
F(x)\geq k\|x\|...
- 5,405
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68
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Fundamental regions in convex programming
In linear programming, the fundamental regions are polyhedra, since those are the intersection of half-spaces defined by linear inequalities. In semidefinite programming, the fundamental regions are ...
- 13.9k
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281
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Examples of geodesically convex functions on (subsets of) $GL_n (\mathbb{R})$
I'm reading this paper [1]. To get some intuition on what a geodesically convex function is like, I'm trying to find some example in $GL_n(\mathbb{R})^+$ that is geodesically convex (near $I$).
Let me ...
1
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147
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Convergence of infinite linear programming
Suppose we have the following linear program (LP1),
$$\min_{f \in \mathcal{C}} \int_{\mathbb{R}} f(t) \cos(2 \pi x_0 t) dt \\ \text{subject to } \int_{\mathbb{R}}f(t)dt = 1 \\\forall x\in [0,1]: \int_{...
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163
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Maximizing variance of bounded random variable through convex optimization
I am interested in maximizing the variance of a random variable $X$ supported on $[0,1]$. Formally,
$$\max_{P_X: X \in [0,1]} {\rm Var}(X),$$
where $P_X$ is a distribution of $X$. This question is ...
- 671
1
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82
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What is the relation between different generalizations of linear programming?
Linear programming subsumed by each of
Semidefinite programming (SDP)
Convex programming (CXP)
SOS programming (SSP)
Is there any relation between each pair in the three?
Are all three equivalent in ...
- 1,826
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40
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Minimax theorems in nonconvex setting
Let $X$ be a topological space, $Z$ be a compact convex subset of $\mathbb R^m$, and let $f:X \times Z \to \mathbb R$ be a continuous function (w.r.t the product topology on $X \times Z$).
Question. ...
- 6,853
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78
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Reference for the algorithm to find the intersection between a subspace and positive orthant
I came across this algorithm, in this question Algorithm for the intersection of a vector subspace with a cone of non-negative vectors ;
Is there any reference for the algorithm described in the ...
1
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0
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67
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What are the corners of this polytope?
Let $f$ be a non-negative function on the positive integers such that $f(s+t)\geq f(s) + f(t)$ for all $s,t\in\mathbb{Z}^+$. Consider the polytope consisting of all $x\in \mathbb{R}^n$ such that $$\...
- 11
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103
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strict convexity of the Legendre-Fenchel transform
Let $d$ be a positive integer.
Let $L:\mathbb{R}^d\to\mathbb{R}$ be a differentiable function with continuous derivatives.
Assume that the Legendre-Fenchel transform of $L$ exists everywhere, is ...
- 11
1
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139
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Subgradient chain rule
Suppose $$F:\mathbb{R}^n \to \mathbb{R},\; F(x)=\mathrm{max}_\mathrm{eig}(C-\mbox{diag}(x)).$$
I am trying to find a subgradient of $F$ at $x_0$. A subgradient of $\mathrm{max}_\mathrm{eig}$ is given ...
- 11
1
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0
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150
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Minimax optimization of diagonal entries of function of matrix
Let $\mathbf{A}$ and $\mathbf{U}$ be arbitrary complex $M\times N$ and $N\times M$ matrices, respectively. Let denote superscript $(\cdot)^{\dagger}$ and $(\cdot)^{\mathrm{H}}$ as pseudo-inverse and ...
- 287
1
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116
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Showing existence of a solution to an underdetermined system of equations with non-negativity constraints
Let $K$ be a positive integer, let $p\in (0,1)$, and let $\{W(k,i),W^B(k,i), \varphi_k(i)\}_{1\leq i\leq k\leq K}$ be variables.
I need to prove that there exists a solution to the following system ...
- 63
1
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79
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Minimization of a smooth integral functional over a closed convex set
Let $(E,\mathcal E,\mu)$ be a probability space, $I$ be a finite nonempty set, $\gamma:(E\times I)^2\to[0,\infty)$ be measurable, $$F_1(g,w):=\sum_{i\in I}\int\mu({\rm d}x)w_i(x)g(x)\sum_{j\in I}\int\...
- 167
1
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282
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total unimodularity of a matrix
Let G be the node-arc incidence matrix of a given directed network (rows of $G$ correspond to nodes and its columns correspond to arcs). Let $B_1,\dots, B_K$ denote a partition of the nodes of the ...
- 393
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167
views
Gradient formula for Clarke's generalized gradient on a general Banach space
In Theorem 10.27 of the book Functional Analysis, Calculus of Variations and Optimal Control, there is the following gradient formula:
($\operatorname{co}$ deotes the convex hull).
Is there an ...
- 167
1
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answers
35
views
Convex optimization problem of determining waiting times
A subproblem of something I'm solving is to determine the optimal intermediate waiting times in a sequence of $n$ jobs. Each job $i$ has an associated waiting time $w_i$ followed by its processing ...
- 11
1
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163
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Can we reduce the maximization of this integral to the maximization of the integrand?
I would like to know whether we are able to reduce the following optimization problem to the pointwise optimization of the integrand (or how we can solve it otherwise): Maximize $$\sum_{i\in I}\sum_{j\...
- 167
1
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0
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188
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Solution to a Strongly Convex Non-smooth Minimization Problem involving an L1 Norm
Let $X \in \mathbb{R}^{n \times d}, w \in \mathbb{R}^d, y \in \{\pm 1\}^{n}, \alpha \in [0,1], \lambda \in \mathbb{R}$. I have an expression that looks as follows
$\frac{1}{2}\|Xw -y \|_{2}^2 + \...
- 11
1
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0
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25
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Weird subspace/equality-constrained LP problem/variant of change-making problem
Assume that we have a set, $\mathscr{R}$ containing $m$-dimensional vectors. Solve
$$\sum_{i=1}^n c_i\leq\delta$$
$$\text{subject to } \sum_{i=1}^n r_i c_i=x^\prime \text{ for all }x^\prime$$
where
$0\...
- 11
1
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0
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208
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Maximum theorem with linear constraints. On parametric continuity of in optimization
Given
\begin{align}
s(\theta)= &\text{arg min}( g( \boldsymbol{x}) ) \\
\text{subject to }& \boldsymbol{A}(\theta) \boldsymbol{x} = \boldsymbol{b}(\theta) \\
&c_1 \le x_i \le c_2 , ...
- 88
1
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43
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How do I find the Rockafellar-Fenchel dual derivation of reguarlized objective function?
I'm stuck finding the dual of this optimization problem:
$$ \max_{\tilde{u},\tilde{v}} \int \left( \frac{1}{2}\left\|x\right\|^2 + \tilde{u}(x)\right) \, ds(x) + \int \left( \frac{1}{2}\left\|y\right\|...
- 383
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36
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Linear programming with a convergent coefficient
The following linear programming problem
$x_n = \arg\min c_n'x \mbox{ subject to } Ax<b$
has a changing coefficient $c_n$. We have that $c_n\rightarrow c_*$. What happens to the solution $x_n$? ...
- 19
1
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answers
126
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Mixed integer formulation of union of polytopes?
Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...
- 1,826
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99
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Finding a point on a convex set
Given a compact bounded convex set $\mathcal C\subseteq\mathbb R^n$ given by $t$ hyperplane inequalities I want to find a point $u\in\mathcal C$ such that for all $v\in\mathcal C$
a convex relation $...
- 1,826
1
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0
answers
24
views
Simple monotonicity property for coordinate descent and linear objective functions
Let $S \subset \mathbb{R}^n$ satisfy $0\leq x_1\leq\dots\leq x_n$ for all $\mathbf{x}\in S$, among other (possibly nonconvex) constraints, and suppose in addition that $\sum_{i=1}^n x_i \geq 1$ for ...
- 4,049
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0
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37
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Fast certficate of negativity for objective value of mixed-integer linear program
Let $c \in \mathbb R^n$, $A \in \mathbb R^{m \times n}$, $b \in \mathbb R^m$, and $I \subseteq \{1,2,\ldots,n\}$. Consider the Mixed integer linear program (MILP)
$$
\begin{split}
f^* = &\max \; ...
- 6,853
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0
answers
163
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Properties of vector combinations in the non-negative orthant
Given a vector $x \in \mathbb{R}^{n}_{0+}$ such that $x = \sum^{k}_{i=1} \alpha_{i}v_{i}$, the vectors $(v_{1},...,v_{k}) \in \mathbb{R}^{n}_{0+}$ are an independent set, $k < n$, and $\alpha_{i} &...
1
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0
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102
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When are quadratic integer programs "easy to solve"?
Let $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I want to maximize $f$ on $N$. $f$ has the following form
$$
f(n) = \sum_i A_i n_i -\sum_i \sum_{j\neq i} B_{ij} (n_i-n_j)^...
- 379
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138
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Matrix completion in $2\times2$ case by nuclear norm minimization to guarantee rank $1$?
Does fixing diagonal entries and minimizing nuclear norm under weighted sum of entries conditions produce a rank $1$ matrix? I think the answer for this is no.
At least could it be true in $2\times2$ ...
- 13.9k
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389
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The perturbation of a convex function can also be convex?
$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...
- 263
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70
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Solve simple stochastic variational inequality
Let $Z$ be a random variable with finite mean $\mathbb E[Z]$ and let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a convex l.s.c function which is differentiable at $z=1$ with gradient $\...
- 6,853
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74
views
How to minimize n-polytope's bounding box with linear transformation?
I am working on an exact algorithm for integer linear programming for my master's thesis:
$Ax\leq b, x \in \mathbb{Z}^n$
$cx\rightarrow min$
For my idea to work out, I need a guarantee that n-...
- 11
1
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answers
59
views
Lower semicontinuity of proximal gradient descent sequence
I'm using an alternate optimization scheme to minimize a function $F(a,b,c) = G(a,b,c) + H(c)$ where $G$ is continuous and differentiable, $H$ is not differentiable but lower semi-continuous and ...
- 11
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266
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Relative interior of a normal cone at a face of a convex polytope?
Suppose $A$ is a nonempty convex polytope in $\mathbb{R}^n$. Suppose $F$ is a face of $A$.
Consider the normal cone of $A$ at $F$:
$C_A(F)=\{v\in\mathbb{R}^n:v\cdot x\geq v\cdot y\ \forall\ x\in ...
- 159
1
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0
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66
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On number of solutions by simplex and number of solutions in total in a linear optimization problem?
This is more of a clarification query.
Mizuno http://www2.ims.nus.edu.sg/Programs/012opti/files/talk_mizuno1.pdf says if we give a linear optimization problem
$$\max c'x$$
$$Ax\leq b$$
where $A\in\...
- 13.9k
1
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0
answers
62
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LP Constraints for Bridgeless Cactus Graphs
When trying to determine the optimal bridgeless spanning cactus graph of a weighted, symmetric graph, I got stuck.
What I do not know how to capture, is
the variable number and sizes of the cycles
...
- 13.2k
1
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0
answers
60
views
On the defect of a flow network
This problem in graph theory was actually motivated by some problems in Theory of Fractals.
To formulate the problem I need to recall some definitions related to flow network.
A flow network is a ...
- 41.8k
1
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0
answers
86
views
Infinite system of equations with finitely many constraints
During my research I have stumbled upon the following issue concerning infinite systems of linear equations. I do not have much practice in such settings, so I am asking you whether the following ...
- 1,151
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0
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53
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Projecting two convex polyhedra onto their intersection
Suppose we are given two convex polyhedra $\mathcal{C}_1, \mathcal{C}_2 \subset \mathbb{R}^n$ with non-empty intersection $\mathcal{C}_1 \cap \mathcal{C}_2 \neq \emptyset$.
For the orthogonal ...
- 337
1
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0
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44
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In convex optimization we know that the optimum solution is on which hyper plane
We have a standard linear program, I mean a set of inequalities $c_i^Tx\leq b_i$ where $i\in \{1,\ldots ,k\}$ and we want to find $max\{c^Ty| y\in \{\cap \{x|c_i^Tx\leq b_i\}\}$. I put some condition ...
- 19
1
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0
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384
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Complexity of conic optimization problems
I am interested in bounding the computational complexity of the interior points method for solving a generic conic problem of the form
\begin{equation}
\min_x \left\{ c^T x : \mathcal{A}x-B\in\mathbf{...
- 345
1
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0
answers
105
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How is the minimax oracle used to find the oracle complexity of projected subgradient?
I am going through a set of blog posts on the complexity of projected gradient method.
https://blogs.princeton.edu/imabandit/2013/03/15/orf523-oracle-complexity-large-scale-optimization/ defines the ...
- 277
1
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0
answers
70
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Lipschitz of the minimum curve of a convex function of two variables
Let $f(x,y)$ be a function in two real variables. Assume that $f$ is strictly convex by which I mean $f(x,y) - c(|x|^2 + |y|^2)$ is convex for some $c>0$.
Assume for each $x$ there is a unique $...
- 280
1
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0
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261
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Prove that the following set of triples forms a convex polytope
Take $a,\,b,\,c,\,d \in \mathbb R_+$ such that $a+b+c+d=1$. Define:
\begin{equation}
x_1 = \min(a+b,\,c+d)\,,\qquad x_2 = \min(a+c,\,b+d)\,,\qquad x_3 = \min(a+d,\,b+c)\;.
\end{equation}
I would like ...
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