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

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Express a second-order cone (SOC) inequality as a linear matrix inequality (LMI)

For $y \in \mathbf{R}^n$ and $t \in \mathbf{R}$, show that: $$||y||_2 \leq t ~~\iff~~ F(y) \succeq 0$$ Where $\text{I}$ is the $n \times n$ identity matrix, and $$F(y) = \begin{pmatrix} t ...
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29 views

does constant convexity assures global minimum

I have the following question: Consider a function $f:R^n \longrightarrow R$, s.t.: there is a point $x_0 \in R^n$ s.t. $\frac{\partial f}{\partial x^k} =0$ $\forall k$. the hessian matrix ...
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162 views

Linear - Quadratic optimization for system of objectives

I have two distinct data sets, $\{x^{\mu},J^{\mu}\}$, $\mu=1,\ldots,n$ and $\{x^{\nu},V^{\nu}\}$, $\nu=1,\ldots,m$ that also include uncertainties $\delta J^{\mu}$ and $\delta V^{\nu}$. In these I fit ...
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37 views

Why is one of the KKT conditions the same as one of the constraints?

I'm working through an SVM tutorial (from Andrew Ng Stanford course notes). In the brief coverage of Lagrange duality. The primal optimization problem is stated $$ \min_{w} \theta_{\mathcal{P}}(w) = ...
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40 views

Constrainted optimization involving logarithms

The problem is to minimize $ f(x_1, x_2 ,x_3, x_4):= - \Big[ \log ({\frac{1}{4} + x_1}) + \log ({\frac{1}{2} + x_2})+ \log ({\frac{1}{5} + x_3})+ \log ({\frac{3}{4} + x_4}) \big]$ such that ...
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15 views

Deriving Dual Averaging from (Sub)gradient Descent

Here the presenter tries to derive a simple Dual Averaging from (sub)gradient descent. I have a little problems understanding the steps. (Sub)gradient descent: Loop through: $$ x_{k+1} = x_k - t_k ...
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17 views

Convergence analysis of gradient descent method

From the following: Convex Optimization (S. Boyd) p.467 Content: We will see that the gradient method does in fact require a large number of iterations when the Hessian of $f$, ...
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41 views

Is inequality $tr(A^{-1^T} B) tr(A^T B^{-1}) \leq constant$ correct?

I have the following optimization problem \begin{align} \min_{A} &tr(A^{-1^T} B)\cr \text{subject to} &x^T A x > 0 \cr & A_{ii}=1 \end{align} where $A$ and $B$ are some positive ...
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41 views

Why is this function, related to SVM derivation, non-convex?

I'm working through a support vector machines tutorial. In eventually deriving the solvable objective function, the following objective function (to be maximized) was proposed, but dismissed as ...
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35 views

Is convex or non convex function?$J(u,c)=\int K(x).u.(f(x)-c)^2dx$

I have a function such as $$J(u,c)=\int K(x).u.(f(x)-c)^2dx$$ where $f(x):\Omega \to R$; c is constant; $0 \le u \le 1$; and K(.) is gaussian kernel. My question is that : Is J convex or non-convex ...
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Inversion of a matrix in a system of linear inequalities

I would like to know if someone knows sufficient conditions on $A\in\mathbb{R}^{n\times n}$ and $b\in\mathbb{R}^{n}$ such that for all $x\in\mathbb{R}^{n}$: $$Ax\leq b \Rightarrow x\leq A^{-1}b \text{ ...
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34 views

How to understand a proposition of subgradient

The question is from the following: Convex Optimization Algorithm (p.512)----- Prof. Bertsekas Let $f: R^n \rightarrow (-\infty, \infty]$ be a proper convex function. For every $x \in ...
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59 views

Calculating the distance function and projection

Let $\Omega=\{x\in\mathbb R^n\mid\langle a,x\rangle=b\}$. We define the distance function and projection as follows $$d(x;\Omega)=\inf\{||x-\omega||\mid\omega\in\Omega\}$$ ...
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23 views

On the solutions of a system of inequalities avoiding Helly's theorem

Let $a_1,b_1,\cdots,a_4,b_4\in\mathbb{R},r_1,\cdots,r_4\in(0,+\infty)$. Show that, if $\not\exists (x,y)\in\mathbb{R}^2$ such that $$ \begin{cases} (x-a_1)^2+(y-b_1)^2\le r_1\\ ...
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25 views

Supporting lines of closed Jordan curve

Given a simple closed Jordan (i.e. continuous) curve $\gamma:[a,b]\to\mathbb{R}^2,\ \gamma([a,b])=C_\gamma$, how can I prove that $D_\gamma$ (the set of all interior points of $\gamma$ toghether with ...
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28 views

Is this floor function/problem quasiconvex?

I am trying to study an optimisation problem under constraints. The point is that all my constraints are linear as well as all terms of my objective function except one. This guy : $$ \alpha^{\lfloor ...
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14 views

Concave-Convex Decomposition of a continuous function

In this paper http://www.stat.ucla.edu/~yuille/pubs/optimize_papers/cccp_nips01.pdf they have a theorem that says that a twice continuous differentiable function (an energy function, they say) with ...
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49 views

Gradient in mirror descent

Mirror descent can be in general written as \begin{equation*} \nabla\Phi(x_{t+1})=\nabla\Phi(x_t)-\lambda_t\nabla f(x_t), \end{equation*} where $f$ is the objective function and $\Phi$ is a convex ...
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52 views

Dual residual for linearized ADMM

I am using linearized ADMM for a problem with a (non-smooth) convex loss function $f(x)$, and a hard constraint $x \in E$, where $E$ is an ellipsoid in $R^d$. I have encoded the hard constraint as $A ...
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2answers
55 views

Is it convex function?

I have a function and I don't know it is whether convex or non-convex: $$J(c,\alpha)=\int_\Omega ( \alpha c-I(x))^2u \, dx+ \|\alpha\|^2$$ where $0 \le u \le 1$, $I(x): \Omega \to R$, $c$ is constant ...
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60 views

Algorithms For Large-Scale $\ell_{\infty}$ Minimization

The general problem I want to solve is well studied: $$ \min_x \Vert Ax\Vert_\infty \;\;\; \mathrm{s.t.} \;\;\; Bx=c, $$ which is equivalent to the following linear program: $$ \min_{t,x} \, t ...
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MATLAB: minimize function using x value from previous iteration

I'm trying to develop an algorithm for a proximal point method defined as: $$ \underset{x \in \rm I\!R^n}{\arg\min} f(x) + \lambda g(x) $$ where f(x) is a convex and coercive function and also ...
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73 views

proving that $\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$

In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that ...
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1answer
36 views

any good way to approximate this non-convex function with convex function?

There is a non-convex constraint in my optimization problem, which is given by $\displaystyle -xy\log\left(1+\frac{z}{xy}\right)$. Obviously, it is neither convex or concave. Is there any good convex ...
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17 views

Help required in solving the lagrangian dual?

I'm trying to write the Lagrangian dual to the following problem \begin{align*} (P) \quad \min\;&\text{Trace}(CG)\\ \text{s.t.}\;&G \succcurlyeq 0\\ & G_{i,i}=I_d (i=1,..,M+1)\end{align*} ...
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2answers
229 views

Proving that $\text{ri rge}\,A=\text{ri conv rge}\,A$

"If $A:\mathbb R^n\rightrightarrows\mathbb R^n$ is maximal monotone,then $\text{ri rge}\,A$ is convex". This is a proposition in auslender's book about the asymptotic cones. We can prove that ...
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54 views

Prove $||\lambda x_1 + (1-\lambda) x_2 - y|| \leq ||x_1 - y||$

Assume we have have $3$ points $x_1, x_2$ and $y$ and $||x_1-y||=||x_2-y||$. How do we prove that the distance between $y$ and the convex combination of $x_1$ and $x_2$ is smaller than that between ...
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Analysis of iterative optimization methods using lyapunov analysis

In analysis of iterative methods, is it possible that we have to use two time-lagged version of the time-varying system to analyze its convergence? (that is, we construct the evolution of x^k, ...
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81 views

Is there any way to make the following function convex?

I need to find optimal lagrangian multiplier vectors for a quadratic programming problem subject to three quadratic equality constraints and several other linear inequality constraints. I would like ...
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84 views

Proving $x^4$ is strictly convex

I'm not sure how to prove $f(x) = x^4$ is strictly convex using just the definition of strict convexity: $$f((1-t)x+ty) < (1-t)f(x)+tf(y)$$ for $0<t<1$. Is this just an algebra slog? If so, I ...
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How to prove $f(x_1,\ldots,x_n) = \sum x_i\ln x_i - (\sum x_i )\ln(\sum x_i)$ is convex on R++ [duplicate]

How can i prove $f(x_1,\ldots,x_n) = \sum x_i\ln x_i - (\sum x_i )\ln(\sum x_i)$ is convex on R++
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71 views

How to compute primal variable based on dual variables and their multipliers

I edited this question based on information I got from comments. Assume we have an optimization problem (primal problem). we solve it's dual using some kind of primal-dual interior point solver. So, ...
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tangent cone to the set

I'm supposed to solve this problem: Let us consider the set $M=\{(x, \sin{x}):x\in\mathbb{R}\}\cup\{\big(\cos(x)-1,x\big):x\in\mathbb{R}\}$ The question is to find the tangent cone to the set $M$ in ...
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53 views

Karush-Kuhn-Tucker conditions for non-linear optimalization

I have the following problem: solve the local conditions (KKT) and find ALL optimal solutions: $$\min f(x,y)$$ subject to $$g(x,y)\le 0$$ $$x\geq0, y\in\mathbb{R}$$ I have some questions to this ...
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1answer
19 views

Gradient of squared distance to a convex set

I have the following problem: Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$, $f(x)=(\operatorname{dist}(x,D))^2$ where $D$ is a convex, close set in $\mathbb{R}^n$. Prove that $f$ is convex and ...
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1answer
14 views

Under what hypotheses is a solution to the Lagrangian multiplier equations automatically a global minimum?

Suppose we are minimizing a function $f(x_1,...,x_n)$ under the conditions $g_1(x_1,...,x_n) = g_2(x_1,...,x_n) = 0$. Under what hypotheses is a solution to the Lagrangian multiplier equations ...
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35 views

Intuition between convex function and convex set

In my text (Luenberger), there is a proposition about convex set Prop: let $f$ be convex function on convex set $\Omega$. The set $\Gamma_c = \{{x: x\in \Omega, f(x) \leq c}\}$ is convex ...
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36 views

Optimization of sum of logs

I have an optimization problem of the form $$\operatorname*{argmax}_{\mathbf{w}} \sum_i \log(1 + \mathbf{w} \cdot \mathbf{k_i})$$ given some set of vectors, $\mathbf{ \{k_i\} }$. I have tried both ...
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89 views

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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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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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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39 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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1answer
23 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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1answer
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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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61 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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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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1answer
32 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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1answer
23 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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30 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 ...