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

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Minimize a convex function over a convex cone

I want to minimize a strictly convex function over a convex cone, where the number of parameter is the same as the sample size. Does the Newton-type algorithm have a global (or local) convergence ...
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Multiplicative version of convex hull

The convex hull of a finite set of points, $(x_i,y_i) \in \mathbb{R_+}^2$ ($i=1,...,n$), is defined as: $$\left\{(\sum_{i=1}^{n} \alpha_i x_i,\sum_{i=1}^{n} \alpha_i y_i) \mathrel{\Bigg|} (\forall i: ...
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17 views

Faster Algorithms for Convex Hulls

I was interested in the following: Given two polyhedra $P_1, P_2$ specified in the form: $$ P_1 = {x : A_1x \le b_1 } $$ $$ P_2 = {x : A_2x \le b_2 } $$ Whereas $ x \in R^n$ and $b_1, b_2$ are ...
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26 views

Convex optimization when Hessian is non-invertible

1) Are there any extensions to Newton's method for finding minimum of a convex function when the Hessian is singular ? (I have all positive eigenvalues in the Hessian except one which is zero) I ...
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21 views

In what ways would a course on convex optimization be useful in game theory?

From talking to several other people in the past, it seems that convex optimization is really a tiny subset of game theory in that it only models the behavior of one single player and does not take ...
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20 views

Submodular function, square of which is also submodular?

A Submodular function $ f:2^E \rightarrow R $ is a function that satisfies the following two equivalent definitions: for every $ S,T\subseteq E: f(S) + f(T) \geq f(S\cup T)+f(S\cap T) $ for every $ ...
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1answer
24 views

about scaling property of proximal operator

If the proximal operator of $f(x)$ is $\text{prox}_{\lambda f}(x)$, what about $cf(x)$ and $f(cx)$, c is a scalar. For example, If $f(x) = ||x||_{1}$, $x \in \mathbb{R}^{n}$, how about the proximal ...
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26 views

Maximum volume inscribed ellipsoid inside nonconvex polyhedron

In Convex Optimization (Boyd, Vandenberghe), an algorithm for finding the maximum volume inscribed ellipsoid inside a convex polyhedron is given on p. 414 (8.4.2 Maximum volume inscribed ellipsoid) [I ...
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31 views

Dual problem of piecewise linear function

I would like to see the geometric interpretation of the relationship between the primal problem and the dual problem on the $x,y$-plane. So I am looking at an example of minimizing the maximum of some ...
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Increasing a singular value [on hold]

Can any one tell me the effect of increasing one singular value (say 10 times ) larger than others.Whether it has any importance in optimization Problems .
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18 views

Partitioning in convex problem (variables in two subsets)

Consider the following problem from textbook Convex Optimization Algorithm p.10: \begin{equation} \begin{aligned} &{\text{min}} & & F(x)+G(y)\\ ...
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Are these two optimization problems equivalent to each other?

Let $\mathbf{x}=[x_1,\ldots,x_K]^T$. For a fixed vector $\mathbf{a}$, I have the following optimization problem : \begin{array}{rl} \min \limits_{\mathbf{x}} & | \mathbf{a}^T \mathbf{x} | \\ ...
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452 views

Convex Optimization of quadratic function with inequality constraints

How would I solve the following problem? $$\min_{x\in\mathbb{R}^n} x^T A x$$ subject to the constraints $$x_i\geq 1,\,i=1,\dots,n,$$ where A is positive semidefinite and symmetric. Is it possible to ...
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61 views

Is the minimum of a parametric convex function convex again?

Let $I$ and $J$ be compact intervals. Let $f:I\times J\to\mathbb R$ be differentiable and strictly convex. Is the function $g:I\to\mathbb R$ defined by $$ g(x) = \min_{y\in J} f(x,y) $$ ...
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1answer
32 views

Finding extreme point of a set determined by two planes in $\mathbb R^3$

Problem asks to find a extreme point the set $\{(x,y,z) \mid x-2y \leq 3 , 2y+3z \geq 4 \}$. But I don't think it has a extreme point, because it is intersection of two hyper planes in 3D, which ...
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17 views

Show that the set $\{y_1a_1+y_2a_2: -1\le y_1,y_2\le 1\}$ is a polyhedron

Show that the set is a polyhedron and express it in the form: $S = \{Ax\leq b, Fx = g\}$, $S=\{y_1a_1+y_2a_2 | -1\leq y_1\leq 1, -1\leq y_2\leq 1\}$ where $a_1,a_2\in\mathbb{R}^n$ My attempt: A ...
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1answer
45 views

Can a convex function have local maxima?

I have read that a convex function can have local maxima. It seems that this must happen on the boundary of the domain, otherwise there should be a region in which the function is concave. Is this ...
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27 views

Reason for use $L^2$-Norm instead of $L^1$-Norm in Optimization [closed]

In optimization we use $\min\; \Vert Ax-b\Vert_{2}^2$ instead of $\min\; \Vert Ax-b\Vert_{2}$ because second is not differentiable. But I am looking for a clean and mathematical reason for this. And ...
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27 views

Biconjugate of a nonconvex function

Is biconjugate of a non-convex function, the tightest lower bound on that function? If yes why?
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how to get orthogonal rank 1 approximations?

The situation: I have $k$ matrices $A_i$, which are all real and of size $m\times n$. Now I would like to find the matrices $\tilde{A}_i$ of $A_i$ so that 1) $\tilde{A}_i$ is of rank 1 (thus a rank 1 ...
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451 views

Using gradient descent and Newton's method combined

I have this function $f(\mathrm{X})$ where $\mathrm{X=A+B+C}$ where $\mathrm{A}$ is a diagonal element with variable $a$ on its diagonal. $\mathrm{B}$ is another diagonal matrix with variable $b$ on ...
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Joint Convexity Proof

Let $x$ be $n \times 1$ vector and $Y$ be $n \times n$ matrix. Prove that $f(x,Y) = x'Y^{-1}x$ is jointly convex in $x$ and $Y$ when $Y \succ 0$.
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Reference Request: Soft handed text on duality theory?

Can anyone recommend a text on duality theory which includes basic formulation of the primal and dual formulation and some introduction to minimax problems? Preferably having some computation in ...
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1answer
39 views

Fenchel Duality in Prof. Bertsekas' lecture

Please see this link, p.39-41 (sufficient to answer my question), before (1.47) for detailed. For convenience, the relevant part is shown as: I am confused in two things: The ...
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1answer
62 views

proximal operator of weighted L1 norm

I hope to solve this problem. $$\min \quad \left\| CX \right\|_{1} $$ $$ \text{s.t.}\quad AX=b, X >0 $$ where $C \in \mathbb{R}^{m \times m}$, $X \in \mathbb{R}^{m \times n}$, $A \in ...
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1answer
53 views

Optimum is achieved when both variables are equal

Consider the problem $\max_{y,z:\|y\|_\infty,\|z\|_\infty \leq 1}y^TBz$, where $B$ is symmetric, positive semidefinite, $y,z\in \mathbb{R}^n$, $\|z\|_\infty=\max_{i\in\{1,\ldots,n\}}|z_i|$. It turns ...
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1answer
483 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 ...
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1answer
441 views

relation between size of matrix and condition number

I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) ...
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574 views

invex functions and their usefulness?

An invex function $f$ is a differentiable function from $\Bbb R^n$ to $\Bbb R$ that for some function $\eta : \Bbb R^n \times \Bbb R^n \to \Bbb R^n$ satisfies for all $x, u$, $f(x) - f(u) \geq \eta(x, ...
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35 views

Finding a polynomial approximation of a PDF

I would like to find a polynomial $P(x)=\sum_{d=1}^D P_dx^d$ of degree $D$, where its derivative is larger than or equal to a given pdf $f(x)$ in $[0,1-\epsilon]$, for any $\epsilon>0$. Note that ...
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Minimizing the sum of the $4^\text{th}$ power of a matrix entries.

Consider a real $n\times n$ matrix $X$. Suppose I would like to minimize the sum of the squares of its entries as a penalty term in some convex minimization. I can write the term using the Frobenius ...
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Intuition behind accelerated first-order methods

$\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ Suppose that we want to solve the following convex optimization problem: $\min_{x \in \mathbb{R}^n} g(x) + ...
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Extreme pts of a polyhedral feasible set

Consider a linear program $\min \{c^T x:Px=q,x\geq 0 \}$, where $P \in \mathbb{R}^{m \times n}$. $x\geq 0$ means each component of $x_i$ of x is nonnegative. The feasible set is $\{x:Px=q,x\geq ...
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1answer
51 views

Conjugacy relation in the primal and dual problem

The following is my derivation in the Conjugacy relation in the primal and dual problem. I am shaky in it; so hope for some advices. Consider the following problem, $f(x),g(x)$ are convex ...
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21 views

Some problems in finding conjugate function

Ask the following fundamental problems: How to derive the conjugate function of $g(y)$ if given "$\underset{y \geq 0}{\text{sup}}\{g(y)-y^Tx\}$"? My attempt is as following: \begin{align*} ...
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133 views

Gradient mapping minimization problem

I am trying to solve this particular problem which can be found in this paper: $$\underset{y\in\Delta_n}{\text{argmin}}\{\langle\nabla f(x),y-x\rangle+\dfrac{1}{2}L\|y-x\|_1^2\}$$ I tried formulating ...
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49 views

How can I solve a nonlinear optimization problem where constraint contains exponential term?

I have the optimization problem as below: $\hspace{10mm} \text{Maximize} \sum_{k} \alpha_k {R}_k $ $ \hspace{10mm}\text{subjcet to:} $ $ \hspace{10mm}\exp \left[ - (2^{{R}_k } -1) \left( ...
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17 views

Optimization problem: solving one implies solving the reverse?

I am looking to solve an optimazation problem $Maximize_{x} [A(x)]$ s.t. $B(x)\geq B_0$, where $B_0$ is a constant. If I solve this problem (i.e, finding the optimal $x^*$ that optmize while ...
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1answer
10 views

Conditions of convergence of stochastic subgradient algorithm

It is well known that for appropriate step size, $E[g^t] \in \partial f(x^t)$ is sufficient conditions for this subgradient algorithm to converge. What I'm wondering is whether the requirement has to ...
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18 views

Condition for product of increasing and decreasing functions to be quasiconcave?

Is there any condition for product of increasing and decreasing functions to be quasiconcave? More specifically, I am having in mind a condition for $F(x)\cdot(1-G(x))$ to be quasi concave where $F$ ...
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1answer
90 views

Trace minimization subject to diagonal constraints

Problem Revisited - Edited for conciseness: We are given two set of data points X [$p \times n$] and Y [$q \times n$]. Let us assume $X = \hat{X} + \tilde{X}$ and $Y = \hat{Y} + \tilde{Y}$ I am ...
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100 views

Showing convexity of a function with the restriction over an arbitrary line

Let $f : \mathbb{R}^n → \mathbb{R}_∞$ be a function and let $C ⊂ dom f$ be a convex set. $$**Part I**$$ Prove that $f$ is a convex function if and only if $f$ is convex over every line $L_{v,x_0}$ ...
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The normal cone at the optimum of unconstrained convex optimization is always $\{0\}$?

I am studying the convex optimization: $\min_x f(x)$, where $x$ is a vector of $p$ elements. In the book CONVEX ANALYSIS AND NONLINEAR OPTIMIZATION Theory and Examples BORWEIN, section 2.1, I ...
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choosing gradienet flows to meet specific requirements, mainly remaining within a convex set

In this paper: http://arxiv.org/pdf/1308.5376v1.pdf, a set of conditions governing the choice of vector field are given. Let $\Delta^n$ be the closed unit simplex in dimension $n$, then for every ...
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Optimize $\max _{x_1,x_2,…,x_N} N , \text{ s.t.} \sum_{i=1}^N f(x_i) \le a$

$Is there general theory for solving optimization problem of the following kind \begin{align} &\max _{x_1,x_2,...,x_N} N \\ \text{ s.t.}& \sum_{i=1}^N f(x_i) \le a\\ &\sum_{i=1}^N g(x_i) ...
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44 views

What's the best way to optimize this energy function, and is it convex?

I have an energy function $E({\bf y})=||\,g({\bf Ay+c})-{\bf d}\,||^2_2 + ||\,{\bf y-e}\,||^2_2 + \alpha\,|{\bf y}|_1$ I need to minimize this with respect to $\bf y$, all other variables being ...
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41 views

What is the relation between these two definitions of an ellipsoid

There are two definitions of an ellipsoid in Boyd's book (Convex Optimization) $E = \{ x | (x-x_c)^T P (x-x_c) \leq 1 \}$ In the above, P is a positive semi definite matrix. $ E=\{ x_c+Au |\; ...
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109 views

how to check an optimization function is convex or not

This is the sparse coding optimization function: $\operatorname*{argmin}_{B, \alpha} \sum_j \| \bf{x}_j - B\bf{\alpha}_j \|_2^2 + \lambda\sum_j |\bf{\alpha}_j|_1$ I read in the literature that this ...
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16 views

Is there any relationship between the rank of mode-k unfolding of tensor X and the rank of X?

I want to solve the following optimization problem: $$\min \|Ax-b\|_2^2 + rank(X)\ \ (*)$$ where $X$ is a three order tensor, $x$ is vec($X$). Many work replace the model as the following one: ...
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28 views

Test for convexity

Consider the online learning setting where instantaneous loss is given by \begin{equation} \ell_t(f_t;(\mathbf{x}_t,y_t))=\max \left( {0,\left( \left( {\frac{N}{P}} \right){I_{(y_t = 1)}} + \left( ...