# Tagged Questions

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

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### Optimization of the function of two variables

I have two functions $f(x,y)$ and $g(x,y)$. I want to minimize the sum of these functions w.r.t $x,y \in (0,1)$. I know that for fixed values of $x$, $f(.,y)$ is a decreasing function while $g(.,y)$ ...
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### Sum of convex and decreasing function

I have a sum of decreasing function and a convex function over some domain. Can I say that the sum is also a convex function (i.e. there exists a unique minimum)?
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### Solve the closed form solution for argmax of $x^Ty - x^T\ln(x)$

Let $y \in \mathbb{R}^n, \ln(x) = \begin{bmatrix} \ln(x_1) \\ \vdots \\ \ln(x_n) \end{bmatrix} \in \mathbb{R}^n$ Show that $$x^* = \text{argmax}_{x \in \mathcal{D}} \quad x^Ty - x^T\ln(x)$$ ...
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### Solving $I^* = \arg\min_{I'} \left( \|\phi_\ell(I) - \phi_\ell(I')\|_2^2 + R(I') \right)$ with gradient descent

I am trying to create the results from this a paper that is trying to understand the types of features a convolutional neural network is learning to recognize. I don't think understanding ...
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### Benders decomposition Master Problem

I am currently working on implementation of Bender's Decomposition for MIP. I am looking at the simplest model \begin{split} \min_{x,y} &\; c^Tx + f(y)\\ s.t. & \; Ax + Dy \ge ...
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### Interpolating polynomial such that it is convex in specified region

The problem I have is that I have data at two points $x_1,x_2$ and $x_2>x_1>0$. At these two points, I know that the function $f$ has values $f(x_1)$ and $f(x_2)$ respectively. It is also ...
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### On accelerated Proximal Gradient Methods

I am working on accelerated optimization scheme, which unified in the paper by Paul Tseng, "On Accelerated Proximal Gradient Methods for Convex-Concave Optimization". But unfortunately, it is ...
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### Is convexity the most general dividing line between “easy” and “hard” optimization problems?

Just got started with Boyd's Convex Optimization. It's great stuff and I see how it directly subsumes the all-important linear programming class of models. However, it seems that if a problem is non-...
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### Does projected gradint descent(pgd) results in the same minimizer as the one given by unconstrained gd and projected back on the constrained set?

For $f: \mathbb{R}^n \mapsto \mathbb{R}$ with $f(x) < \infty,\;\forall x \in \mathbb{R}^n$ and for convenience let's assume $f$ is continuously differentiable. Suppose we are trying to solve the ...
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### Finding minimizer from different order

Let a nonnegative function $f(x,y)$: $\mathbb R^2\to \mathbb R$ be second order continuous differentiable. We also know that $f$ is not convex in its two arguments, but only separately in each of them....
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### Is generalized mean convex / concave?

The generalized mean can be given using the following equation: $M_p(x_1, \dots, x_n) = (\frac{1}{n}\sum_i x_i^p)^{1/p}$ Is it convex /concave when $p<1$ ?
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### Conjugate of difference of convex functions

I am reading through this tutorial on DC programming and the author makes a startling claim without proof: If $g$ and $h$ are two lower semi-continuous convex function, then the conjugate function of ...
I am studying logistic regression and in my book it says that using Hessian we can show that $f(w) = \sum_{i=1}^N (\frac{1}{1+e^{-w^Tx_i}} - t_i)$ is not convex. Both $x$ and $t$ are N-Vectors, and $... 1answer 25 views ### Connection between complementarity problem and optimization problem? I do not understand the connection between complementarity problems and optimization problems. I have tried to look at other definitions for complementarity problem to see if that would help me with ... 2answers 26 views ### Variational Inequalities - What excatly does the definition say? Why are they useful? I am having issues understanding the definition of variational inequalities. We have the following definition: Given a set$X \subset \mathcal{R}^n$and a mapping$F: X \rightarrow \mathcal{R}^n$a ... 2answers 100 views ### Decrease in the size of gradient in gradient descent Gradient descent reduces the value of the objective function in each iteration. This is repeated until convergence happens. The question is if the norm of gradient has to decrease as well in every ... 1answer 38 views ### Minimizing the Frobenius norm with linear inequality constraints How to solve the following system for$\mathbf{C}$and$\mathbf{a}$:$\min\|\mathbf{X-XC} \,\mbox{diag} (\mathbf{a})\|_F^2$subject to$\mathbf{c}_{ik}\geq 0$,$1^T \mathbf{c}_k = 1$and$1-\delta\...
Given the following optimization problem (Orthogonal Projection):  {\mathcal{P}}_{\mathcal{T}} \left( x \right) = \arg \min _{y \in \mathcal{T} } \left\{ \frac{1}{2} {\left\| x - y \right\|}^{2} \...