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

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Convex analysis books and self study.

I have taken some courses in Convex optimization. Now I would like to know a little bit more about the pure mathematical side. Is there any good books in convex analysis? I have read and worked with ...
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Subgradient of convex minimization duality

$$\min(f_0(x))$$ $$\text{s.t. }f_i(x) \le y_i \forall i, i = 1 ,\ldots, m$$ $$f_i : \text{convex};\quad x : \text{variable}$$ It is also considered that $g(y)$ is the optimal value of the problem ...
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205 views

Optimization of relative entropy

Wondering if my following question is an application of information theory: Lets say we have a factory and ship boxes of stuff outside. If a competitor stands outside my factory, observes the stream ...
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Need advice: what should be my next step?

I am dealing with a quite algebraic question and I arrived at some good point. I had $2$ equations with $2$ unknowns and I was able to eleminate one of the variables. My final equation still seems ...
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How to find accuracy of Matlab's quadprog solver?

I have solved with quadprog from Matlab a strong convex quadratic problem given as $$ f(x) = x^TQx + c^Tx$$ with constrains $$ Cx \leq b.$$ Now the output of quadprog is: Minimum found that ...
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On a version of gradient descent

I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has coordinate-wise Lipschitz constant $M$ (meaning that for all ...
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103 views

Primal-Dual pair in SDP

Let's we have a primal model like $\max~~ x + \langle I, Z \rangle$ $s.t. ~~~Ax + y I - Z \preceq B$ $~~~~~~~~~Z \succeq 0, ~X \geq 0, ~~y ~free$ where $A, B \in {\mathbb R^{n \times n}}$. The ...
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When can the optimal value of a SDP be achieved?

Looking at semidefinite programs, are there any sufficient conditions for the solvability (i.e. the optimal value can be achieved, that is infimum=minimum)? Obviously if the problem is unbounded, the ...
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If this problem is not unbounded, what's wrong in this dual derivation?

In a paper with 100 citation, Robust Support Vector Machine Training via Convex outlier Ablation, a convex relaxation is used. In this paper, a form of robust svm proposed: \begin{align} \min_{0\leq ...
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Is minmax equivalent to maxmin?

More precisely, problem $1$ is as follows: \begin{eqnarray} &\max_{1{\le}i{\le}N}\min_{[\gamma_m^i]}\left[\lambda_i - \sum_{m=1}^M\phi_m\gamma_m^iF_m^i\right] \\ &\mbox{subject to} ...
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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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Are these equivalent definitions of faces of convex sets?

I consulted several books and found that the definitions of faces and exposed faces of convex sets are a bit messy. Many books just treat them as the same stuff. In book "foundations of ...
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Online stochastic convex optimization.

I need to find/approximate the argument that minimizes a stochastic convex function $F(\theta, Z)$: $$ {\arg\min_{\theta}} E_{Z}[ F(\theta, Z) ]$$ Where $Z$ is some random variable (we could assume ...
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63 views

Proving or disproving concavity of a function

I want to prove that the following function is concave (as a part of another proof). $$f(p) = \max_{\begin{matrix}x,y\\0\le x \le 1\\0\le y \le 1 \\ x * y = p\end{matrix}} \lambda h(x) + ...
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54 views

The difference between affine set and affine hull

According to the definition of affine hull and affine set. $$aff [C] = [\theta_1x_1+...+\theta_nx_n|x_1,...x_n \in C, ...
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66 views

Normalize gradient

I want to minimize a function $f \, : \, \mathbb{R}^{N} \, \longrightarrow \, \mathbb{R}$ (with $N \in \mathbb{N}^{\ast}$. In my problem, $N = 315$). I know that $f$ is differentiable on ...
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Strong duality: When does the optimal primal variable coincide with the primal variable giving the dual function.

I'm considering the inequality-constrained optimization problem of finding $$ x^{\star} = \arg \min_{x} f(x) \;\; \text{s.t.} \;\; h(x) \le 0 $$ which is assumed to have a unique minimizer. The ...
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The importance of the full-row-rank assumption for the simplex method

Consider a linear programming model in the usual form ready for applying the simplex method. I understand that having the constraint equations' coefficient matrix $A$ be of full row rank means not ...
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Does convexity of a function guarantee tractability of finding its minimum?

Formulating a problem as a convex optimization problem usually implicitly considered to imply being able to find global minimizer of the objective. My question is that if it is true or not. ...
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the dual of the dual is the primal?

Consider a convex optimization problem (call it $P$). Consider its dual (call it $D$). Is it true that the dual of $D$ is $P$? For linear programming, it is true. I'd just like to know under which ...
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On the duality gap for quasiconvex optimisation problems

This stack exchange question got me thinking about quasiconvex analysis. Given a compact,convex subset $X\subset \mathbb{R}^n$ and a quasiconvex function $f:X\rightarrow \mathbb{R}$ Define the ...
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How to show that a function is piecewise linear

Let z(t) = min $(c+t d)^T x$ s.t $Ax <= b$ Show that Z(t) is a concave, piecewise linear function of t. I'm really not sure how to even start proving this, I would really ...
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Global optimum of sum of convex functions

Take two real differentiable convex functions, $f_1$ and $f_2$, defined on the unit interval $[0; 1]$. I want to find the global optimum of: $\min_{x \in [0;1]} af_1(x)+bf_2(x)$, for given $a, b \in ...
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How to calculate the maximal ellipsoid in a given polyhedron

I am faced with the problem of finding the ellipsoid $B$ ($B$ is a symmetric positive definite matrix) of maximal volume within a convex set $C$ given as a set of linear inequalities $C=\{x| a_i^T x ...
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Convex minimization over the Unit Simplex

I have a simple (few variables), continuous, twice differentiable convex function that I wish to minimize over the unit simplex. In other words, $\min. f(\mathbf{x})$, $\text{s.t. } \mathbf{0} \preceq ...
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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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Least expensive way to “walk” with a convex potential

Let $V(x)$ be a non-decreasing, convex function on $\mathbb{R}^+$. Fix some positive integers $L$ and $k \leq L$. Let $A_{L,k}$ be the space whose elements are sequences of positive integers $r= ...
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Accelerated Gradient Descent V.S Nonlinear Conjugate Gradient Descent

Let's consider smooth and convex minimization problem, i.e. $min f(x)$, where $f$ is not necessarily a quadratic function. If measured by iterations, Accelerated Gradient Descend (AGD) has ...
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How does one evaluate the derivative of a matrix with a tensor $\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}$?

I am stuck on the following: $$\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}=\text{ ?}$$ with $A$ a $d\times d^2$ matrix, $\mathrm{Id}$ the identity matrix of $d\times d$ ...
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how to find the edges emanating from a given vertex in a polyhedron

Suppose my polyhedron $P$ is defined as $P={ x\in \mathbb{R}^n \mid Ax=b, x\geq0 }$ I have $x_0$, which is a vertex of $P$. How to find the edges emanating from $x_0$? In other words, I want to find ...
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solving the primal problem via dual

On pp. 248 of Boyd and Vandenberghe: suppose 1) strong duality holds, 2) the dual optimal is attained at $(\lambda^*, \nu^*)$, 3) the dual function $L(x, \lambda^*, \nu^*)$ has the unique minimizer ...
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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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the continuity of argmin on convex funtion

Define $$x'=\text{argmin}_{x_1}f(x_1,\lambda),$$ where $f$ is a strictly convex function on $x_1$ and $\lambda$. I would like to ask if there is any theorem about the continuity of $x'$ w.r.t ...
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Log concavity/convexity of a determinant

I was wondering if anyone would be able to help me determine whether the following quantity is log concave or not with respect to $\alpha$? $$\left[\det(\textbf Y^\top \textbf P \textbf G \textbf ...
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Maximin problem as LP?

Consider the following setting. Let $A\in \mathbb{R}^{3 \times m}$ and $B\in \mathbb{R}^{m\times 3}$ be two matrices such that each of their columns must add up to a given $c\in \mathbb{R}$. Denote by ...
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What decides the structure of the dual variables taken in designing min-max type combinatorial optimization algorithms?

There are a bunch of combinatorial optimization problems like min cost flows and min weight perfect matchings that invoke duality and complimentary slackness to improve the primal feasible solution. ...
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Dual Decomposition with multiple coupling constraints

This is probably a a simple question, but have been stuck on this for a while and unable to figure out my issue from the standard Boyd/Vandhenbergen decomposition references. I am interested in dual ...
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Is optimal solution to dual not unique if optimal solution to the primal is degenerate?

If optimal solution to the primal is degenerate, does it necessarily follow that optimal solution to dual not unique? That is, is uniqueness an unnecessary assumption? Spin-off from here. In my ...
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Strong duality for nonconvex quadratic program (with multiple constraints)

Consider the following optimization \begin{eqnarray} P_1: \quad &\underset{x\in\mathbb{C}^N}{\mathrm{minimize}}&\; f_0(x) \\ &\mathrm{subject\;to}&\; f_i(x) \leq 0, i=1,\ldots,m \\ ...
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discrete nonlinear convex optimization relaxation over a dense set

Be a discrete nonlinear convex optimization problem $P$ \begin{align} \underset{x\in \mathrm{C}^n}{\mathrm{min}} \ \ \ f(x) \\ Ax=b \\ c \leq x \leq d \end{align} $C$ is a dense in $F$. Is solving ...
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Can we ever have E(argmin(f)) = argmin(E(f))?

Consider a parametric real-valued function $f_{\boldsymbol{\alpha}}:\ \mathbb D^N \rightarrow\mathbb R$ whose parameters $\boldsymbol\alpha$ vary according to some distribution $\psi$, and $\mathbb D$ ...
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proving that the ewma is convex

Hi: I know it's true but I don't know how to prove that exponentially weighted moving average when view ed as a function of $\lambda$, is strictly convex. The exponentially weighted moving average ...
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Effective convexity criterion for the finite point set in $\mathbb{R}^3$

I need to find effective convexity criterion for the finite point set. Below there is description of what is meant by "effective" criterion. Definition. Let $M = \{A_{1}, \ldots, A_{n}\}$ be the ...
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Affine functions as equality constraints in convex optimization problems

I am studying on an introduction to convex optimization problems. When defining a convex optimization problem, we have a convex object function, $f(x)$, a set of convex functions $g_i(x)$ where the ...
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Complexity analysis of convex optimization problem

I am studying an optimization problem \begin{equation} \mathbf{x}^*=\text{argmax}\quad\sum_{d=1}^{D}\log(\mathbf{a}_d^T\mathbf{x}+b)+\mathbf{c}_d^T\mathbf{x}+f_d\\ \text{subject to}\quad ...
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Solve the linear program

Please help to solve this problem. I am new to this type of problems and any help will be greatly appreciated $$\text{ Minimize } 7x-5y+3z$$ $$\text{ Such that } \ \ \ 0 ≤ x ≤ 6 , -2 ≤ y ≤ 7 , -4 ...
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How to prove the convexity of the logrithmic gamma function?

Here's what I did: $$\Gamma'(z)=\int_0^\infty \log(t)e^{-t}t^{z-1}dt$$ $$\Gamma''(z)=\int_0^\infty ...
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Strictly convex self-concordant function

Some definitions: A function $f:R^n\rightarrow R$ is convex[strictly convex] if for every $\lambda\in[0,1]$ [$\lambda\in(0,1)$] and for every $x,y$ [$x\neq y$] in $R^n$ we have $f(\lambda ...
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Semidefinite Program formulation

I have the following problem and would like to formulate that as an SDP. I am not sure how to approach this : A set $S$ is given such that : $$ S = \{P \in R^{n \times m} : ||p_i - c_i|| \leq d_i ...
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Uniqueness of the solution

We know that 1) Minimise of a convex function the unique solution exists 2) Maximise of a concave function the unique solution exists How about 1) Minimise of a strictly convex function? 2) ...