Convex Optimization is a special case of mathematical optimization. It includes Linear Programming and least-squares.

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How to project gradient vector to subspace defined by linear constraints

I have the following set of linear constraints: $$\begin {align}\textbf{y}^T\textbf {x} &= 0 \\ \textbf {0} &\leq\textbf {x} \leq C\cdot\textbf {1},\end {align}$$ where $\textbf {y} \in ...
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24 views

Is the normalized duality mapping symmetric?

In the area of functional analysis, nonlinear operator theory, the normalized duality mapping on a Banach space $X$ is a set valued map from $J:X\rightarrow 2^{X^*}$ given by \begin{align} ...
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21 views

Product of linear and convex function

More specific, how many maxima are there for product of these two functions: $ f(x) = ax + b $, and $ a > 0 $ $ g(x) $ is (strongly) decreasing convex function, $ \lim_{x\rightarrow\infty} g(x) = ...
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37 views

A maximization problem within the simplex

Let $\lambda_i$ be an ordered list of $N$ positive numbers, $\lambda_1<\lambda_2<\dots<\lambda_N$. I'm looking for the extrema of the function $$ f=\left(\sum_{i=1}^N p_i \lambda_i ...
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Example of convex subset (unbounded) with $\text {rec} (C) = {0}$

Example of convex subset (unbounded) with $\text {rec} (C) = {0}$ I've proved that for a bounded convex subset $C$ it always holds that $\text {rec} (C) = {0}$. However, now I'm looking for an ...
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21 views

How can a second-order cone problem be expressed as a conic problem?

I realize that a second-order cone is a cone, and thus an SOCP is a type of conic problem. However, to me it doesn't seem so apparent, looking at their equations. Could someone explain how one could ...
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21 views

Find the edges of a polyhedron P.

Given the polyhedron $P = \{v \in \mathbb R^2 \mid Av \le b\}$ with $A = \begin{bmatrix} -1 & -1 \\ 2 & -1 \\ -1 & 2 \\ 1 & 2 \end{bmatrix}$ and $b = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 2 ...
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58 views

How to check if given polyhedron is empty

Consider a polyhedron specified as following set of equalities and inequalities $$ \begin{aligned} &\mathbf{A}\mathbf{x} = \mathbf{b},\\ &\mathbf{x} \geqslant \mathbf{0}. \end{aligned} $$ Are ...
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10 views

Is every optimization problem with a piece-wise affine objective function the dual of some differentiable problem?

It is well known that a problem can have a $C^1$ objective function and a convex feasible set, while the dual problem can be piece-wise $C^1$ only. So I'm wondering - if you have a piece-wise affine, ...
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22 views

Proximal Operator of $\ell_{\infty,1}$ norm of a matrix

How can I calculate the proximal operator of mixed norm $\ell_{\infty,1}$ for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_{\infty,1} + \frac{1}{2\tau} ||X-Y||_F^2$ where ...
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17 views

Relaxing the elements of a matrix

I try to understand a specific part of the paper "Consistent shape maps via semidefinite programming", where a binary symmetric Input matrix $X^{in}$ is given with $X^{in} \in \{0,1\}^{nm \times nm}$ ...
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20 views

Do quasi-Newton methods check the second-order optimality condition?

I have a practical question about quasi-Newton methods. In quasi-Newton methods, Hessian matrix is approximated. It seems to be impossible for them to check the second optimality condition. In ...
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37 views

Weighted least squares with nuclear norm minimizaiton, how to optimize?

Nuclear norm minimization is very popularization and formulation is least squares term with nuclear norm term as following, $$\min\limits_{X} ...
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21 views

Sion Minmax theorem for integral operators

Suppose $f, g\in S=L^p([0,1],\Sigma,\mu,[0,1])$. The objective $L:S\times S\to R$ is given by $$L(f,g)= \int f (h-g) d\mu, $$ where $h\in S$ is fixed. Could we apply Sion Minmax theorem to conclude ...
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40 views

Why a convex cone cannot have more than one extreme point?

The way I define an extreme point is : A point which cannot be defined as a convex combination of two distinct points. I'm not able to extend this and show why a convex cone cannot have more than ...
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30 views

Convexity of |y-X'w|^2 given that inverse of X does not exist

Can we say anything about the convexity of the $|y-X^Tw|^2$ if we know that inverse of $X$ does not exist? Hessian is $2XX^T$. Given that X is not invertible, can we conclude that $XX^T$ is ...
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Convergence of backtracking and gradient descent.

I am thinking a bit about the following exercise: Let $f(x) = x_1^2 + x_2^2$ with dom $f = \{ (x_1,x_2):x_1 > 0 \}$. The optimal value of this problem is $p^* =1$, but it is never attained since ...
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20 views

Check if convex polygon is completely contained completely within another convex polygon.

How can I determine if a convex polygon is completely contained within another convex polygon where speed is critical? I've thought about doing this, which will only use inequalities: pcp = ...
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12 views

If a quadratic form $f$ takes the minimum on a triangle in a vertex, what can I say about min of $f$ on edges of a subdivision?

Let $f(x)=x^2+y^2$ be the Euclidean square-norm and $A,B,C\in\mathbb{R}^2$ be vertices of a triangle $\Delta$ such that $f$ takes the maximum on $\Delta$ in $C$, the minimum in $A$ and takes the ...
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33 views

Using semidefinite programming to solve the following problem

I am struggling with the following problem, and wonder is SDP can help: $$\mathrm{maximize\ } \alpha_{10}+\alpha_5+(\alpha_2+\alpha_8)/2 \mathrm{\ subjected\ to\ } \mathrm{T_1}\succeq0, ...
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11 views

Convex combination and convex set

From where does $tx + (1-t)x'$ originate from? I am selfstudying an economists book, and this is popping up all of a sudden. I get that it's a line between $x$ and $x'$, but why? And is $tx' + (1-t)x$ ...
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23 views

Solving Constrained Least-Squares

I need to solve a constrained least-squares (LS) problem as follows $min_X \text{ } ||Y-AX||_F^2$ $s.t. \text{ } {X\in \chi}$ where $A\in R^{n\times m}$, $(n\ge m)$ , $X\in R^{m\times k}$ and ...
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25 views

How can I compute fast the minimum of a linear plus Kulback-Leibler on the unit simplex?

Given $a, x^0 \in \mathbb{R}^n$ I wish to compute $$\min_{x \in \Delta_n} a^t x + \sum_{i=1}^n x_i\log(x_i/x^0_i) - x_i +x^0_i $$ where $\Delta_n$ is the unit simplex $\{x \in \mathbb{R}^n \mid ...
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32 views

For a non-convex function f, how to find a function g such that $g\circ f$ is strictly convex?

The following function $f(x)={1\over (1+e^{-x})}$ is non-convex but $\ln(f(x))$ is convex. Given a non-convex function $f$, can we find a function $g$ such that $g\circ f$ is strictly convex? If yes, ...
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20 views

How to convert the following optimization problem to quadratic program?

Given positive constants $C$ and $\epsilon$ and points $\{ (x^i,b_i)\} _{i=1}^I \subset \mathbb{R}^{n+1}$, how can we rewrite the following optimization problem as minimizing a convex quadratic ...
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19 views

Book on duallity and sensitivity in nonlinear optimization

I am looking for a recommended book on duallity and sensitivity in nonlinear optimization, as duallity and sensitivity is a well studied topic in LP , I am struggeling to find books in this subject ...
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20 views

Continuity of optimisation problem

Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$ I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and ...
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Convergence rate - Convex optimization

What is the best known algorithm in terms of convergence rate for unconstrained convex optimization and under what assumptions? $\min_{x} f(x)$ where $f(x)$ is a given twice differentiable convex ...
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20 views

Show the Gini Coefficient is Quasiconvex

The Gini-coefficient is defined as $$ G(x) = \sum_{i = 1}^n \frac{i}{n} - \sum_{j=1}^{i} \frac{x_{(j)}}{\mathbb{1}^{T}x}, $$ where $x_{i} $ is nonnegative numbers with positive sum. $x_{(j)}$ denotes ...
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38 views

Augmented Lagrangian Method for Inequality Constraints

Augmented Lagrangian Method can be used with inequality constraints. The question is how. One approach (according to Numerical Optimization Book by Nocedal and Wright; page 522), is linearly ...
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31 views

Formulate an optmization problem as a convex optmization problem

Let $P$ be a polyhedron, i.e. $P = \{ x \in \mathbb{R}^{n}\, |\,\, a_{i}^{T}x \leq b_{i} \}$. Define $R$ as the rectangle given by $\{ x \in \mathbb{R}^{n}\, \mid\, \, l \preceq x \preceq u \}$. Find ...
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64 views

intuition behind subspace of $R^n$

Hi: I've been reading an optimization text by Charles Byrne, "A First Course In Optimization". I'm currently going through the chapter where he explains things about convex sets and convex functions ...
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34 views

strong convex implies exp-concave

Prove that if f is strong convex (for some m>0) $\mbox(\nabla f(\mathbf{x})-\nabla f(\mathbf{y}))^{T}(\mathbf{x}-\mathbf{y})\geq m||\mathbf{x}-\mathbf{y}||_{2}^{2} $ then f is also ...
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34 views

proof that length of difference of projections implies equality of length of normals

Hi: I'm reading a book on optimization and there is an interesting stated theorem but I don't know how to prove it. Notation: Let $P_{c}(x)$ denote the projection of onto a convex set c which is a ...
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48 views

$\arg\max$ of an increasing function grows as the region grows.

$f_1,\dots,f_N:\mathbb{R}^+\rightarrow\mathbb{R}^+$ are strictly increasing, bounded functions whose derivatives monotonically decrease to $0$ as their argument increases. (Picture the shape of the ...
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41 views

Non-convex function with global minimum [duplicate]

I am working on a complicated objective function which I suppose is not convex. But when I use a global optimization tool that can find all its local minimums, it will always converge to the same ...
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29 views

How can I minimize a quadratic on the unit simplex?

How can I compute $$ \min_{x \in \Delta_n} \frac{1}{2}\lVert Bx\rVert^2 + x^tAy$$ with $x \in \mathbb{R}^n, y \in \mathbb{R}^m, A_{m \times n}$, $B_{n \times n}$ where $\Delta_n$ is the unit simplex ...
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Proximal operator of spectral norm of a matrix

How can I calculate the proximal operator of spectral norm for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_2 + \frac{1}{2\tau} ||X-Y||_F^2$ I understand that the proximal ...
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20 views

How can I solve $\min \{ \langle A(x),y\rangle + f(y) \text{ s.t. } y \in S^n, \operatorname{tr}(y) =1, y \geq 0\}$?

I'm trying to solve the problem $$\min \{ \langle A(x),y\rangle + f(y) \mid y \in S^m, \operatorname{tr}(y) =1, y \geq 0\}$$ where $x \in \mathbb{R}^n$, $y \in S^M$, that is, it's a symmetric $m$ by ...
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Krein Milman Property

If a closed bounded (not compact) set $X$ in a Banach space $B$ (like $L^1$) has extreme point(s), must the max of a linear functional defined on $X$ occur at one of them? I suppose it depends on $B$. ...
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How to solve this entropy optimization problem with gradient projection method?

The problem is defined as $$ \min_{w} = \sum_{i=1}^{n} \sum_{j=1}^{n}\left\{ \left(\sum_{k=1}^{3}w_kP_{ij}^{(k)}\right) \log \left(\sum_{k=1}^{3}w_kP_{ij}^{(k)}\right) \right\} + \gamma \|w\|_2^2\\ $$ ...
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Minimizing a function known to have a unique local and global minimum

Quasi-convex functions are a class of functions known to have a unique local and global minimum, which can minimized over convex sets using numerical methods with convergence guarantees. A function is ...
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40 views

Proximal operator for the nuclear norm of Hankel (x)

I have a problem in hand for which I need to compute the proximal operator of the composite function $||Hankel(x)||_{nuc}$ where $x \in R^N$ and $||.||_{nuc}$ denotes the matrix nuclear norm. For a ...
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54 views

linear approximation with respect to L1 norm

I am trying to solve this problem: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize $\int_0^1|e^x-\alpha-\beta x| dx$ any hints how to proceed
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Find an optimal solution for $\min_{x} F(x)$ analytically

I want to find an analytical solution (exact/closed-form) for $x$ of the following minimization problem: $$\min_{x} b x \left[e^\left(\frac{a}{x}\right)-1\right]+d (1-c-x) \left[e^ ...
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29 views

How to solve the least square with $L_2$ norm constraint directly?

I answered the question Why are additional constraint and penalty term equivalent in ridge regression? earlier, but I myself still have some questions on it. To solve \begin{align} \min_{\beta} ...
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41 views

Constructing a quasiconvex function

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$ f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, ...
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47 views

Is the optimum of this problem unique?

I have a convex optimisation problem: $$\min_{x_i} \sum_{i=1}^N a_i \exp(-x_i/b_i)\quad\text{ s.t. }\sum_{i=1}^N x_i=x\text{ and } x_i\ge 0$$ The KKT conditions are: $$\lambda=\begin{cases} ...
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12 views

Reference material on Alternating Minimization Algorithm

I am looking for some good reference material (book/paper) for learning Alternating Minimization Algorithm. Any recommendation from optimization experts will be much appreciated. Thank you.
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207 views

(Updated) Geometric Illustration of Monotone and Maximal Monotone Maps

I am writing a note about the Monotone and Maximal Monotone maps from the following book http://link.springer.com/book/10.1007%2Fb97594 In this book we read a map ...