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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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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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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46 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 ...
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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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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 ...
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3answers
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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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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 ...
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20 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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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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106 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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29 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: ...
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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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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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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 $n$ ...
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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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31 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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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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54 views

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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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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32 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 + ...
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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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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 ...
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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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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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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 ...
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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} ...
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Optimization by Symmetry?

Let $$f(x,y,a,b) := \frac{xa+yb}{\sqrt{xa^2+yb^2}},$$ where $x,y,a,b$ are all positive. Define $$g(a,b) = \min_{x+y=1,\,x,y\ge 0}f(x,y,a,b).$$ How would one solve for $g(a,b)$? I have solved this by ...
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Using optimization for a logarithmic function

Question: A tangent line is drawn on the graph of $y=\ln x$ for $0\lt x\lt 1$. A right triangle is thus formed in the fourth quadrant. If we regard the area of this triangle has a positive value, ...
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How can we constrain lagrange multipliers in svm dual by adding constraints in primal problem?

Consider svm-dual,i.e., \begin{align} &\text{maximize} \sum_{i=1}^n \alpha_i-\frac{1}{2\lambda} \sum_{i,j=1}^n \alpha_i \alpha_j y_i y_j K(x_i,x_j)\cr &\text{subject to, } 0\leq \alpha_i ...
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Am I headed in the right direction with this area optimization question?

Question: A fence will create a rectangular area with one side being formed by an existing building (and hence, the fence only needs 3 sides). One side will be created using Redwood fencing and the ...
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Is this a Combinatorial Optimization problem with Multiple Constraint Satisfaction?

Given n-dimensional data consisting of over 20000 samples with 200 dimensions, using this as an example: ...
2
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1answer
68 views

Knapsack in graph

This question is from job interview for a software company.   "You are given an undirected connected weighted graph with $n$ nodes. The weight function represents transportation costs. In ...
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37 views

Maximum number of teams of three people such that each team is built in one of two ways

A coach picks team members in two ways:   A. The team of three people should consist of one experienced participant and two newbies. Thus, each experienced participant can share the ...
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Multi objective optimization: Ideal vector

I'm going to consider the two problem distinctly. Now I want to calculate $z_1^{id}$ and $z_2^{id}$ and $x_1^{id}$ and $x_2^{id}$ where $z_1^{id} = min(x)$ $z_2^{id} = max(y)$ $z_1^{id}$ is the ...
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How to transform this problem to a matrix optimization problem?

I wonder how can I loose the following set of equations to an optimization problem ? Suppose given three real vectors $w_0$, $f$ and $\delta$, and a positive entry vector $c$, such that for every ...
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1answer
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Why lower semicontinuity?

I'm reading a proof on the existence of a solution to a minimisation problem, but I'm stuck. I give a brief summary of the arguments up to the point at which I'm stuck(at the yellow box). ...
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49 views

Auction Design : Multiple lots, one win max per bidder, not regret

This is a real life game theory problem. I have to organize an auction. There is a finite number of lots, which are not equivalent. There is a finite number of bidders; the number of bidders is ...
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1answer
56 views

Area of the figure within the circle and outside a polygon

For which values of the parameter $c \in \mathbb{R}$, the area $S$ of the figure $F$, consisting of the points $(x,y)$ such that $$\begin{gathered} \max \{ \left| x \right|,y\} \geqslant 2c \hfill ...
9
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124 views

gradient flow on $SU(n)$

Define the following cost functions $f_1, f_2 :SU(n) \rightarrow \mathbb{R}$ by $f_1(U) = Re \left( \text{Tr}\left(G^{\dagger} U \right) \right)$ and $f_2(U) = \left| \left( \text{Tr}\left(G^{\dagger} ...
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1answer
40 views

An optimization problem, in the form of a word problem,

The manager of a $1000$ seat concert hall knows from experience that all seats will be occupied if the ticket price is $50$ dollars. A market survey indicates that $10$ additional seats will remain ...
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Find the function $\hat{g}$ that maximizes $\int_0^1 x^2g(x)dx$ over the set of all functions that satisfy the following conditions:

Find the function $\hat{g}$ that maximizes $\int_0^1 x^2g(x)dx$ over the set of all functions that satisfy the following conditions: $\int_0^1 |g(x)|^2dx =1$, $\int_0^1 g(x)dx=0$, and ...
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Bivariate optimal density

Consider any feasible $p:[0,1]^2\to [0,1]$ that allows discontinuities and the problem $$\min_{p(.)} \int_0^1\int_0^1 p(x,y)^2 dF(x) dG(y)$$ s.t. $$\int_0^1 p(x,y)dG(y)=k\phantom{0} for \phantom{0} ...