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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How to find the sum of distances so that it is minimal?

Question: $A$ and $B$ are two points on the same side of a line $l$. Denote the orthogonal projections of $A$ and $B$ onto $l$ by $A^\prime$ and $B^\prime$. Suppose that the following distance are ...
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10 views

How close is $\operatorname{argmax}_p E[\log(f(p,\alpha)]$ to $\operatorname{argmax}_p \log(E[f(p,\alpha)])$?

Here $\alpha$ is a random variable and the expectation is taken with respect to that variable. I am wondering if it's the same in any case or there's a theorem quantifying how close both things are. ...
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0answers
32 views

What is the minimizer of the matrix norm and it's significance?

For $M_{n\times n}$ a p.s.d real matrix, if we minimize $||M^{\frac{1}{2}}x||_2$ over $x$ under a linear constraint on $x$ as in $Ax=b$, where $b$ is non-zero. what is the significance of this $x$? ...
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11 views

Is there a minimizer associated with this function in which Hessian is given?

Given a certain function f(x) which we want to minimize the Hessian is found to be: $ \begin{bmatrix} 8&-4\\-4&8 \end{bmatrix} $ And the the point that satisfies the FONC is found to be $ ...
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78 views

Global Minimum of $f(a) = \int _{-\infty}^{\infty} \exp\left(-|x|^a\right)dx, a\in(0,\infty)$

Playing around with the Standard Normal distribution, $\exp\left(-x^2\right)$, I was wondering about generalizing the distribution by parameterizing the $2$ to a variable $a$. After graphing the ...
0
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0answers
18 views

Minimize residual Matrix

I have two Matrices $A$ and $B$ of order $m\times n$ and Matrix $C=A-B$ I want to formulate a optimisation problem such that I get the difference at each element should go be minimum, ie, a null ...
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1answer
40 views

What's wrong in this dual derivation?

I have a function in the form \begin{align} f(q,M)=\sup_{0\leq \alpha \leq 1} -\alpha^T (R\odot M)\alpha+\alpha^Tq \end{align} which is a dual of a minimization problem, where $R$ and $M$ are ...
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35 views

What is the solution of this optimization problem?

I am looking for the solution of this optimization problem: $$ \min_{x} \max_{1 \leq r\leq N-1} \left|\frac{\sin\pi r M x}{\sin\pi r x}\right|^2$$ where $M \ll N$, $x \in \mathbb{R}$, $r \in ...
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0answers
30 views

$\mathbf{A}$ is unimodular $\Rightarrow$ $\mathbf{A}$ has entry in $\{-1, 0, 1\}$?

Is it true that $$\mathbf{A}\;\text{is unimodular}\;\Rightarrow\mathbf{A}\;\text{has entry in}\; \{-1, 0,1\}?$$ Also can an unimodular matrix $\mathbf{A}$ has entry in $\mathbb{R}$?
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1answer
99 views

Minimization of a combinatorial function

The following gamma function depends on the overall sum of $x_n,x_j,x_k$ $$\gamma(X)=\sum_{x_n+x_j+x_k=X}\left [ \left ( \prod_{i=1}^{s}(x_{ni}-1)!C_i^{x_{ni}} \right )\times ...
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0answers
19 views

Optimize to Find the Mahalanobis Distance to Minimize the Term

I have an optimization problem defined as following: Assuming we have a data set $ { \left\{ \left( {x}_{i}, {y}_{i} \right) \right\}}_{i = 1}^{N} $ where $ {x}_{i} \in {\mathbb{R}}^{d} $ and $ ...
0
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0answers
15 views

How do you compute the weighted sum of data points for learning the centers of a hyper basis function network (HBF)?

I was reading the following paper on hyper basis function (HBF) (similar to radial basis function RBF network) and was trying to figure out how one learns the movable centers of the hyper basis ...
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0answers
26 views

Minimizing variance subject to linear inequality

Let A be a $n \times n$ matrix. Where $A$ is a symmetric positive definite matrix. Let $b$ be a vector in $R^n$. $x$ is an unknown vector to be determined. I'm interested to find vector $x$ such ...
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16 views

Maximisation of the distance of particles in a periodic box

Consider $N$ particles in a box of ratio $R=L_x/L_y$, where $L_x$ and $L_y$ are the two sides of the box. The box has periodic boundary conditions. Consider now a state which maximises the distance ...
0
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1answer
27 views

Find the minimum number of tanks to hold the maximum quantity of wines, at each tank maximum possible capacity

My business is in the wine reselling business, and we have this problem I've been trying to solve. We have 50 - 70 types of wine to be stored at any time, and around 500 tanks of various capacity. ...
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17 views

Notion of outer normal cone and supporting cone if $x \in$ relint($C$)

In my lecture we defined the outer normal cone $ N_c(x^*)= \{ c\ \in \mathbb{R^n} : \max\limits_{x \in C} \ \ c^Tx = c^Tx^* \}$ and the supporting cone $S_C(x^*)= \bigcap\limits_{c \in N_c(x^*)} ...
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1answer
39 views

How can I use Banach Contraction Principle to solve $Ax = b$?

Can anyone explain to me how Banach Contraction Principle (fixed point theorem) makes it easier to solve $Ax = b$?
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2answers
45 views

Largest Arithmetic Sum of Relatively Prime Numbers Under 30

Pick however many integers in the range $[1,30]$ (inclusive). The only constraint is that all of these numbers must be relatively prime to each other. What is the largest possible arithmetic sum ...
2
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17 views

EM algorithm with constrained equation

I am reading a paper where author uses EM for the following equation to find the parameters $\theta$(and $\beta$) : $$ J=\sum_m \alpha_{m}\sum_i\sum_j w_{mij}\log\sum_k \theta_{ik}\beta_{mjk} $$ ...
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26 views

How to solve following optimization problem for a classifier using Lagrangian relaxation and subgradient? [closed]

\begin{align} \min &\quad H\\ \mathrm{s.t.} &\quad H \geqslant y_i(Wx_i+b)\\ &\quad y_i(Wx_i+b) \geqslant 1 \end{align} I am unable to get how to solve this optimization problem without ...
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0answers
25 views

How doI solve given linear optimization equation by subgradient method? [closed]

Given the linear program \begin{align} \min &\quad cx\\ \mathrm{s.t.} &\quad Ax\geqslant b\\ &\quad Bx\geqslant d \end{align} The Lagrangian lower bound program would be reduced to: ...
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44 views

Computational complexity of the following quadratic program (QP)

Let $A^TA$ be a $n \times n$ matrix. I have the following quadratic program to solve: \begin{array}{rl} \min \limits_{x} & x^T A^T A x \\ \mbox{subject to} & \sum_{i=1}^{r} x_i =1, ...
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21 views

How to judge the convexity of this function?

$ f(X) = -\log \det(X^TX+I)$, $X \in \mathbb{R}^{n \times n}$, is this function convex or not? Does anybody have an idea about this problem? Thanks.
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2answers
50 views

Minimizing a summation?

I have absolutely no idea how to approach this problem. I've been looking through notes, and I think I missed this when my professor discussed this in class. $$ \text{Consider the data}\\ i\: x_i\: ...
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15 views

Choice of multivariate constrained optimizer

I was trying to find an optimizer-language combination that minimizes a cost function with ~100 variables and box constraints fairly quickly. The function is nonlinear and non-differentiable (at one ...
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4answers
45 views

Maximum area of rectangle circumscribed about another rectangle.

While studying, I came upon this problem: "Find the maximum area of a rectangle circumscribed about a fixed rectangle of length 8 and width 4." I looked at the answer key, which showed that the ...
2
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3answers
39 views

Find maximum and minimum values of an equation on an elipse

I need some help with this. I've been struggling through this last chapter of my Calc III class, and I'm not sure how to do this (although, it doesn't seem like it should be difficult to do) $$ ...
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3answers
44 views

How to find a pareto optimal solution in a smart way (3 variables)

$\max\left( { 3x }_{ 1 }+4{ x }_{ 2 }+2{ x }_{ 3 } \right) $ ${ x }^2_{ 1 }+{ x }^2_{ 2 }+{ x }^2_{ 3 }\le 1 $ ${ x }_{ i }\ge 0 $ I have to find a Pareto Optimal solution, but I can't solve this ...
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10 views

Can Somebody Help Me Find A Certain Paper about Hybrid Proximal Extragradient method for Bregman Functions?

I have read these two papers by Svaiter and Solodov. The first one, published in 1999 (http://pages.cs.wisc.edu/~solodov/solsva99Teps.pdf) presents an error criterion for the hybrid proximal ...
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14 views

How should I determine the optimum input parameters?

I am writing an application to help the user select the best algorithm, from a list of numerous possibilities, to solve a specific signal processing problem. I know the correct answer so I can compute ...
2
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0answers
19 views

Closest line to point after non-linear map

I have a map on a vector space $M(\vec{r})$, defined as below. All components (vectors, matrices, everything) are reals in the unit range $[0,1]$. The map $M(\vec r)$ is defined as the sum of an ...
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2answers
27 views

Minimizing cost for a given volume

288 m3 tank will be made in the form of a rectangular prism. The cost of 1 m2 of top and bottom walls is 40 euros. The cost of 1 m2 of side wall is 30 euros. What should be the edges to be cheap as ...
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3answers
105 views

Which statement “must be false”?

Given a function $f$ continuous on $[-4, 1]$ with its maximum at $(-3, 5)$ and its minimum at $(1/2, -6)$, is it not correct to say that both statements (B) and (D) must be false? (A) The graph of ...
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2answers
27 views

Rewrite matrix equation as a quadratic programming problem

Given real-matrix $X_{n\times p}$ how can the problem of minimizing $Tr(X^TA_{n\times n}X)$ under the constraint $Tr(X^TC)=\phi$ be posed as a standard convex quadratic program given by the form: ...
4
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1answer
86 views

Is this a known result?

I heard the following result and I am wondering if anyone can verify its correctness and also provide a source to cite. If the Lagrangian $L(x,\lambda)$ is convex in $x$ at the optimal Lagrange ...
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2answers
15 views

KKT multipliers sign convention

We all know that if we have an optimization problem of the general form: $$ \min f(x) $$ subject to: $$\begin{align} h(x) &= 0 \\ g(x) &\le 0. \end{align}$$ Then, we have a vector of ...
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27 views

Can this problem be a form of Nonlinear Programming?

How we can reformulate blew problem as a form of nonlinear programming problems? $$ \begin{array}{ll} & \min&\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n} &\\ & ...
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15 views

Jointly learning with dependences

I am looking for some directions/opinions regarding the following problem. I am new to this domain and although I searched in the web, I am totally lost on which direction to take. I have $n$ ...
2
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1answer
38 views

Optimization with Calculus

This one has me bugged big time. An architect needs to design a rectangular room with an area of 89ft^2. What dimensions should he use in order to minimize the perimeter? Round to the nearest ...
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1answer
30 views

Prove the solution $u_i$ of equation $\lambda r_i+\frac{1}{\theta}(u_i-v_i)+\beta \left (\sum_{i=1}^Nu_i-1\right )$

I have an cost function such as $$E(U)=\lambda \sum_{i=1}^{N} \int_{\Omega}r_iu_idx+\frac{1}{2\theta}\sum_{i=1}^{N} \int_{\Omega}(v_i-u_i)^2dx+ \frac{\beta}{2} \int_{\Omega}\left ( ...
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22 views

Transformation between two optimization problems.

Problem $1$ is as follows: \begin{equation} \max_{1{\le}i{\le}N}\min_{\{v_i\}_{i=1}^N\in\textbf{V}_{\gamma}}\left[\lambda_i - v_i\right] \end{equation} Problem $2$ is as follows: \begin{eqnarray} ...
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+50

Derivation of back-propagation equation $\frac{\partial E(\theta)}{\partial W^k}=x*\delta h^k+\tilde{h}^k*\delta y$ for convolutional autoencoders

I was reading the following paper on convolution stacked auto-encoders and they had the following convolution neural network (for auto-encoders, notice I didn't write the offset term [to avoid ...
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1answer
16 views

Optimizing a positive definite quadratic form with inequality constraints.

I have a positive definite, multidimensional quadratic form: $(x-x_o)^t M (x-x_o)$, where the "${}^t$" indicates transpose and $M$ is a positive definite matrix (in fact, it is a multidimensional ...
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31 views

Bilinear Constraint

I would like to formulate the following Optimization problem. My question is focused on the constraint. Given a "typical" objective function, e.g.: $$ \min c^T v $$ s.t. $$ 0 = a_1 v_1 - a_2 v_2 + ...
3
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1answer
30 views

Proof that $\min_{b\in B} u(a,b)\leq \min_{b\in B}\max_{a\in A}u(a,b)$

So I have two finite sets $A,B$ and $u:A\times B\rightarrow \mathbb{R}$ a utility function. I am asked to give a certain proof but I don't need help with the whole thing, I just need help figuring ...
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21 views

Is there a solution/algorithm for a one-directional stable marriage problem?

I'm working on an application to assign summer camp kids to activities. The program input is a list of campers, and a ranked list of each camper's preferred activities. For example, Alice might rank ...
0
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0answers
20 views

Do the higher-order Frechet derivatives can be used in optimization?

I am working on inverse problem in seismics. In our community, we use 1-order and 2-order Frechet derivatives a lot to solve the inverse problems being posed. However, some seismologist verified that ...
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18 views

Newton-Raphson convergence for function $f(\gamma)$ with $\gamma \geq0$ constraint.

I am reading an (engineering) paper that in the part of their solution, They propose a 2-step iterative solution based on Newton-Raphson method for concave function $f$ as below : ...
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1answer
25 views

Game Theory (continuous utility, pure strategy)

I have a game in which two players, 1 and 2, choose a non-negative real number level of effort $e_1,e_2$ respectively. Their cost for this choice is $ce_i$ for $i=1,2$ where $c>0$ is the same for ...
0
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1answer
33 views

Non-vanishing of sub gradient near optimal solution

Consider the non-smooth optimization problem \begin{equation} \min_{x \in \mathbb{R}^n} f(x). \end{equation} To solve the above problem, I am suing subgradient descent \begin{equation} ...