# Tagged Questions

Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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### Can this problem be formulated as an integer programming problem?

I am working on an optimization problem but I am not sure if the problem can be formulated as an integer programming problem. Assume the cost minimization problem for a set of subscribers and ...
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### Optimize wrt a partial matrix?

I have a common optimization problem $$\arg\min_A \text{tr}( A^TWA),$$ where $W$ is a positive semi-definite matrix, and $A$ is the matrix to be optimized. If $A$ is completely unknown, with some ...
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### Maximizing $\frac{\int_r^1xf(x)dx}{2-F(r)}$

Consider a continuous distribution on $(0,1)$ with probability distribution function $f$ and cumulative distribution function $F$. Define $$g(r)=\frac{\int_r^1xf(x)dx}{2-F(r)}$$ and let $r_M\in(0,1)$ ...
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### Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
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### Is Frank Wolfe a descent algorithm?

A colleague was explaining to me that the Frank-Wolfe algorithm is a descent algorithm (i.e. its objective value decreases monotonically at each iteration). However, when I tried simulating it, my ...
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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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### Maximize the number of non zero elements of a product of binary matrices.

I want to find two binary matrices $A$ of size $N \times M$ and $B$ of size $M \times N$ such that: $AB=C$ is a strictly lower-triangular matrix ($j \geq i \implies C_{ij}=0$) The number of ...
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I have an optimization problem of the form \begin{align} \text{maximize}\quad&\sum_{i=1}^{k}\sum_{j=1}^{n}w_{ij}x_{ij}\\\text{s.t.}\quad \quad\quad\,\,& \sum_{i=1}^{k}x_{ij}\leq 1,\;\forall ... 0answers 8 views ### Proximal operator of the sum of two norms I'm a little new to this and not sure how to evaluate the proximal operator in this context, assuming a closed form exists. Calculate \text{prox}_{c,f+g}(v), where f(x) = \frac{1}{2}\|Ax-b\|_2^2... 1answer 67 views ### Team grouping troubles Imagine there are 12 teams, numbered 1 through 12. There are 10 games those teams can compete in, with two teams needed per game. There are 10 rounds, and it is important that after the 10 rounds are ... 0answers 21 views ### Find rotation matrix to match points in parallel projection I am given two sets of 3D points (actually 2D, see below) with corresponding pairs. I am seeking two 3D rotation matrices, such that (only) the X and Y components of the rotated points match best (... 1answer 31 views ### Minima of symmetric polynomials subject to two symmetric constraints The homogeneous symmetric polynomial of degree k in n variables is f_k(x_1,x_2,\dots,x_n) = \sum_{i_1<i_2<\cdots<i_k}x_{i_1}x_{i_2}\cdots x_{i_k}. $$Consider the following ... 0answers 16 views ### Solve \max \mathrm{sum}(AXB \geq \gamma), X \in \{0,1\}^{N \times N} I have a problem to find the best permutation matrix X \in \{0,1\}^{N \times N}, so as to maximize the number of elements in AXB which are above a certain positive number \gamma. In other ... 1answer 957 views ### Why does SVD provide the least squares solution to Ax=b? I am studying the Singular Value Decomposition and its properties. It is widely used in order to solve equations of the form Ax=b. I have seen the following: When we have the equation system Ax=b, ... 2answers 36 views ### Maximizing the sum of the squares of numbers whose sum is constant I wonder how one goes about to find the maximum of \sum v_i^2, the v_i's being positive integers whose sum \sum_i v_i is fixed. 0answers 15 views ### 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 ... 1answer 14 views ### In Constrained Optimization, Restrict Domain to Open Set A\subset\mathbb{R^N}? In constrained optimization and context of economics (e.g. utility function with quantity of goods as arguments subject to wealth), why do textbooks always restrict domain of the objective function ... 0answers 18 views ### Invertibility of bordered Hessian I have an optimization problem: max f(x) s.t. Ax=b, where x \in R^n and b \in R^m, m \le n. I know that f is strictly quasi-concave, and that A has rank m (linearly independent, ... 1answer 456 views ### How to solve for maximum area of a rectangle under a curve? Having trouble with this optimization question and was hoping I could get some help with it. The function of the curve is 8^{-\frac{x}{5}}. I would greatly appreciate a full explanation. I already ... 0answers 15 views ### Solve  \max U = [\sum\limits_{i = 1}^2 {a_i^{{1 \over \sigma }} } \cdot X_i^{{{\sigma - 1} \over \sigma }} ]^{{\sigma \over {\sigma - 1}}}  The problem is$$ \eqalign{ & \max U = \left[\sum\limits_{i = 1}^2 {a_i^{{1 \over \sigma }} } \cdot X_i^{{{\sigma - 1} \over \sigma }} \right]^{{\sigma \over {\sigma - 1}}} \cr & \...
I would like to maximize the term $l_1b_1+l_2b_2+l_3b_3-2$ such that the following conditions hold: $1>l_1>l_2>l_3>0$, $l_1,l_2,l_3 \in \mathbb{Q}$, $b_1,b_2,b_3 \in \mathbb{N}$...