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

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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
16 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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2answers
45 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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0answers
5 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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1answer
18 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
15 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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0answers
14 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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31 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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19 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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2answers
31 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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0answers
8 views

In optimization, does weak duality imply the primal and dual formulations are inverses of each other? [on hold]

Wikipedia states: In mathematical optimization theory, duality means that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem (the duality ...
2
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1answer
27 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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0answers
8 views

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 ...
2
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1answer
16 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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1answer
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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0answers
29 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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1answer
9 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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1answer
19 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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1answer
21 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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1answer
24 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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17 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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17 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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19 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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1answer
28 views

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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1answer
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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1answer
28 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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29 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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1answer
57 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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31 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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1answer
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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52 views

Positive solutions to $A^T A x \geq 0$

Find a positive solution $x$ to the linear inequality $A^T A x \geq 0$. Progress. One special solution is when $A^TA$ is row diagonally dominant, then the column vector $1$ satisfies $A^T A x ...
3
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1answer
46 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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0answers
40 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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0answers
28 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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0answers
30 views

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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0answers
19 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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0answers
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Help understanding the specification of constraints for cvxopt

This is an example from the cvxopt documentation and I am trying to understand how the L2 constraints are specified to the solver. The problem is specified as: ...
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0answers
27 views

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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15 views

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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30 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 ...
2
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1answer
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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0answers
29 views

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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0answers
26 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} ...
2
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1answer
39 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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2answers
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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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0answers
10 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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2answers
196 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 ...
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1answer
23 views

Is this function jointly convex in its variables?

I have a function which I suspect is jointly convex, but have a difficult time proving it, especially since the Hessian is messy. The function is $f(y_i,i=1.2,\ldots,N)=\sum_i l_i w_i + y_i$, where ...
0
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0answers
35 views

Is there a unique saddle value for a convex/concave optimization?

Here is the question: Consider a function of two variables $f(x,y)$ on some compact domain. Let it be convex in $x$ and concave in $y$. Is it true that $f(x,y)$ has a uniqe saddle point ...