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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Pontryagin's Maximum Principle as a sufficient condition?

It is know that Pontryagin's maximum principle provides in general only a necessary condition in the following sense: The ODE system which is known to be solved by the optimal control may have ...
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48 views

How to find the minimum of $f(a_1, b_1, \ldots , a_m, b_m) = \sum_{j=1}^{n} (y_j - \sum_{k=1}^{m} a_kx_j^{b_k})^2$?

$$f(a_1, b_1, \ldots , a_m, b_m) = \sum_{j=1}^{n} (y_j - \sum_{k=1}^m a_kx_j^{b_k})^2$$ $$2m < n$$ $x$ and $y$ are constants, and $a$ and $b$ are variables to find. I took deviation out of it and ...
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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 ...
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1answer
15 views

Extrema function of two variables problem

A rectangular box with a square base is to be constructed from material that costs 9 dollars per $ft^2$ for the bottom, 7 dollars per $ft^2$ for the top, and 4 dollars per $ft^2$ for the sides. Find ...
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Best approximation and an inequality

Let $H$ be a Hilbert space. Let $E\subset H$ and $x\notin E$. Suppose that there exists $y^*\in E$ such that $$\|x-y^*\|=\min_{y\in E}\|x-y\|$$ (i.e., $y^*$ is the best approximant of $x$). I hope ...
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24 views

Gradient of a sum of indicators

Say I have a function $\mathbb R^n \rightarrow \mathbb R$: $$f(w_1,\ldots,w_n) = n^-\sum_{i\in I^-}w_ix_i$$ with fixed $x_i\in\mathbb R$ (data), $I^-$ the set of indexes with negative sum operands ...
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35 views

Dynamic programming recursion

In a book by Wayne Winston for operations research I found this question. Here's how I did it: Let $t$ be the no.of subjects to pass and let h be the no.of hours she has in hand for studying. ...
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14 views

Can a function be both lower (but not upper) semicontinuous and convex?

Is it possible to construct such an example? For example, can a discontinuous function $f : \mathbf{R} \rightarrow \mathbf{R}$ be also convex?
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26 views

Extrema Function of $2$ variables [closed]

I do not know how to set this problem up. Any insight as to how to get the equation would be great. It is John's birthday and his parents want to make him a cake in the shape of a rectangular box. ...
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1answer
18 views

linear programming : Absolute value in constraint in mathematical model

I have a model have an constraint with evaluation of absolute value , a example can be: function objective : $\max \sum(x_i)$ statement: $x_i\geq |(y_i-t_i)|$ for all $i$ but value absolute ...
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Significance of lower semicontinuity in (non-)convex optimisation

In the context of (non-)convex optimisation, what is the reason behind requiring that the objective function be lower semicontinuous? From what I understand, 1) a function is continuous iff it is both ...
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51 views

The maximum of a functional

Is the following statement true or false? $$ \max F \left( \theta \right)= \int_{\rho_{min}}^{\rho_{max}} g \left( \rho \right)\pi\left(\rho,\widehat{\theta \left( \rho \right)} \right)d\rho$$ ...
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19 views

How to efficiently select a subset of elements that maximizes a certain property? (entropy)

I need to select $k$ elements from a pool containing a much larger number $N$ of elements. The selection must be done in a way that a function $h(\{z_{i_1},\ldots,z_{i_k}\})$ is maximized or ...
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17 views

Double integral of a product in calculus of variations

Let's say I have an integral of the form $$ V(u) = \iint\limits_{[0,T]^2}f(x,y)u(x)u(y)\mathrm dx\mathrm dy $$ which I would like to optimize over smooth functions $u$. For the variation I get $$ ...
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21 views

Multivariable gradient descent with approximation of gradinet

This is not a statistics problem I have a vector $$X=[x_1,...,x_{10}]$$ and a cost function $$y=F(X)$$ and my aim in to find the best $X$ to minimize the cost function. It is impossible to ...
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35 views

Optimization involving the length of $x$ as the perimeter of two geometric objects

I am having problems understanding how to solve the following optimization problem: A piece of wire 12 m long is cut into two pieces, the length of the first piece being x m. The first piece is bent ...
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60 views

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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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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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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63 views

Proof of convergence for the proximal point algorithm

I'm trying to come up with a super simple proof of convergence on the proximal point algorithm, which uses the iteration scheme $x^{i+1} = \mathbf{prox}_{tf}(x^i)$ where $f$ is a closed, convex ...
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12 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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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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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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440 views

Convex Optimization of quadratic function with inequality constraints

How would I solve the following problem? $$\min_{x\in\mathbb{R}^n} x^T A x$$ subject to the constraints $$x_i\geq 1,\,i=1,\dots,n,$$ where A is positive semidefinite and symmetric. Is it possible to ...
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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 $M \ll N$, $x \in \mathbb{R}$, $r \in ...
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$\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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834 views

Linearization of a product of two decision variables

I am trying to solve a problem that involves constraints in which products of two decision variables appear. So far, I read that such products can be reformulated to a difference of two quadratic ...
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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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Optimization with a few Variables (AMC 12 question)

In the 2013 AMC 12B, question 17 says: Let $a$,$b$, and $c$ be real numbers such that $a+b+c=2$, and $a^2+b^2+c^2=12$ What is the difference between the maximum and minimum possible values of $c$? ...
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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 ...
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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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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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Image restoration in matlab via PDE toolbox

I want to remove a noise for an image using matlab, when the observed image is $$f=u+v$$ where $u$ is the restored image (is the image i want recovered) and $v$ is the gaussian noise. To restore $u$, ...
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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 ...
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A lower bound on sorting algorithms

I think I have a proof that $n\ln n$ is optimal in the sense that is it a lower bound for sorting algorithms. See here for a list. It must be greater than $n$ as this is too linear, and the $\ln$ ...
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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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146 views

Calculus question with optimization homework

A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (a) How much of the wire should go to the square to maximize the total area ...
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445 views

Using gradient descent and Newton's method combined

I have this function $f(\mathrm{X})$ where $\mathrm{X=A+B+C}$ where $\mathrm{A}$ is a diagonal element with variable $a$ on its diagonal. $\mathrm{B}$ is another diagonal matrix with variable $b$ on ...
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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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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 ...
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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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How to solve $\min \limits_{\mathbf{x}} \| \mathbf{Ax}-\mathbf{b} \|^2$?

Let $\mathbf{x}=[x_1,\ldots,x_K]$. I have the following optimization problem: \begin{array}{rl} \min \limits_{\mathbf{x}} & \| \mathbf{Ax}-\mathbf{b} \|^2 \\ \mbox{s.t.} & x_k\ge 0, \forall ...
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Modelling a warehouse in optimization?

I am trying to model the following rather general optimization problem. Let $p_{t}$ be a given non time series of product prices. These are fixed points $p_{t}$ is not described as a random variable. ...
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Hölder's inequality/Cauchy-Schwarz for Bregman Divergence?

Consider the Bregman divergence. $$ D_F(p, q) = F(p)-F(q)-\langle \nabla F(q), p-q\rangle. $$ And its dual norm: $D_{F*}(p, q) $ where $ F^*(y) = \arg\min_x \left\{ \langle x, y\rangle - F(x) ...
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1answer
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How should I treat a linear relaxation to a rucksack optimization problem?

I am currently studying for an exam in optimization, and in one of the questions the following was mentioned: "The linear relaxation of the problem $(P)$: $z^*=\max ...
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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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482 views

Smooth approximation of maximum using softmax?

Look at the Wiki page for Softmax function (section "Smooth approximation of maximum"): https://en.wikipedia.org/wiki/Softmax_function It is saying that the following is a smooth approximation to the ...
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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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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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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 ...