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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Optimization problem involving step function

I've got to optimize the following function with respect to $\phi$: $q(\phi, x) = \frac{1}{n} \sum_{i=1}^{n}{H(y_i)}$ where $y_i = k - \phi l - x_i$ and $H(.)$ denotes the Heaviside function. $k$ ...
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Integer combination

i want write a module to find the integer combination for a multi variable fomula. For example $8x + 9y \le 124$ The module will return all possible positive integer for $x$ and $y$.Eg. $x=2$, ...
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Rigid-body matching algorithm and clustering algorithm with groups of lines in 3D

I've been struggling with this problem for weeks, and couldn't find an appropriate algorithm to solve it. Could you guys please give me some advices or suggestions in addressing this question. Or if ...
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244 views

Calculate the value of $f(x)$ which minimizes the sum of three functions given a range of $x$

I have the requirements to minimize the following: $$ (f(x)_1 + f(x)_2 + f(x)_3) $$ where: $$ f(x)_1 = y_1 - (\exp(b+m_1) \times x) $$ $$ f(x)_2 = y_2 - (\exp(b+m_2) \times x) $$ $$ f(x)_3 = y_3 - ...
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Solving an optimization problem involving reciprocals

I am trying to solve the following minimization problem, perhaps by getting it into a LP form: Let $u= [u_1, u_2, ...u_N]^T$ a column vector, and $v=[{1\over u_1}, {1 \over u_2}, ...{1 \over u_N}]^T$ ...
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104 views

Hausdorff-like distance between two arrays

Let $(X,d)$ be a metric space and $a,b\in X^n$ be two arrays of elements of $X$. Define $$ \rho(a,b):=\inf\limits_{\sigma\in \Sigma}\sup\limits_{1\leq i\leq n}d(a_i,b_{\sigma(i)}) $$ where the ...
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Maximize the area of the inscribed triangle

Problem Try to determine the maximum area of the inscribed equilateral triangle of a ellipse with semi-major axis $a$ and semi-minor axis $b$. Thoughts Suppose the equilateral triangle is ...
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Universal Correlation measure — ranking correlations

I have time series data of experimental observations for two related processes. I want to measure correlation for use in further analysis. Correlation of the series changes over time and across ...
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731 views

First-order condition for one kind of optimization

I encounter an optimization problem like this: $$\min_{w(x)}{\int {w(x)f(x|e)dx}}$$ subject to $$\int {v(w(x))f(x|e)dx} - g(e)=u$$ $w(x)$ is a function and suppose it has desirable ...
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Root Convergence rate of Iterative Scheme [closed]

I have an iterative sequence for optimizing an EM algorithm based loss function $L(X)$ with $t$ being the iteration number as: $X_t=ABX_{t-1}+CX_{t-1}+X_{t-1}$ where $A$ is a diagonal matrix, $B$ and ...
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289 views

Finding saddle point of a quadratic form

I am trying to find the saddle point of a quadratic form: $$f(\mathbf{x})=\mathbf{x}^\mathrm{T} \mathbf{A}\mathbf{x}+\mathbf{x}^T\mathbf{b}+\mathbf{c}$$ using a minimization/maximization-like ...
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30 views

Order of evaluations

I have a list of objects with $N$-dimensional criteria (actually hotels in my case). Now I'd like to optimize the order of the criteria by which I filter to spent the least time on some evaluation ...
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120 views

Orthogonality, Maximization and Eigen-Solution

I Have read that for a matrix of reals $Y$ and a p.s.d matrix $B$ that the Maximum of $ f(Y)=Tr(Y^TBY)$ subject to $Y^TY = I$ is achieved when $span(Y)$ equals the span of the first $d$ ...
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432 views

Understanding how to state the Karush-Kuhn-Tucker Conditions for a given problem

I'm trying to understand an example given by Nocedal & Wright (1999), pg 329, Example 12.4. According to a definition given earlier in this book: At a feasible point x, the inequality ...
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86 views

$l_1$ norm projection with regularization term

I recently encountered an optimization problem and looking for some technical paper for the same.The problem is give as below, $\min f(x)+\lambda*r(x) $ $\ s.t \ x \geq 0, ||x||_1 = 1$. where $x$ ...
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86 views

Maximization of a sum subject to constraints on 3 resources

This is a generalization of a subproblem from a past programming competition that I had trouble with. Given input $6$ positive integers: $$r_1, r_2, r_3, x_1, x_2, x_3 \in \mathbb{Z^+}$$ ...
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406 views

Determine the nature of a critical point (Lagrange multipliers)

Let $F \colon \mathbb R^2 \to \mathbb R$ be the function $$ F(x,y):=xye^x + ye^y - e^x+1 $$ and denote with $C$ the set of zeroes of $F$, i.e. $C:=\{(x,y) \in \mathbb R^2 : F(x,y)=0\}$. Let ...
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Finding closest vector subject to a constraint

I have the following optimization problem and I don't know how to approach it, I'm not even sure if I'd be able to get a closed form solution: $$\min_b \|d-b\| \\ \text{s.t.} Ab < y $$ I'm trying ...
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138 views

Maximizing Volume [duplicate]

Possible Duplicate: Maximizing volume of a rectangular solid, given surface area Maximize the volume of a rectangular solid, given that the sum of the areas of the six faces is 6a^2 for a ...
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100 views

Software to optimize a quadratic program with quadratic constraints

I'm working in eight dimensions and want to minimize $x^TAx$ under the constraints $x^TBx \geq c$. Unfortunately, A is not positive semidefinite. Worse, I am almost positive that my domain is not ...
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A rectangle is inscribed in a circle of radius 8m [closed]

A rectangle is inscribed in a circle of radius 8m a. Find the dimensions of the rectangle that will maximize the area of the rectangle b. Find the maximum value of the area
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Optimization with non-negativity and norm constraint

I am facing the following optimization problem: $$\min_x w^tx \\ s.t. ||x|| = 1, \forall i: x_i \geq 0 $$ where $w$ and $x$ are real valued vectors. How would I solve this? My background is not ...
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Lower bound of $J=\frac{x^TAx}{x^TBx}$

Consider two symmetric positive semi-definite matrices $A, B \in \mathbb{R}^{n\times n}$. Suppose that $A$ and $B$ have the same null space $\mathcal{N}\subset \mathbb{R}^n$. Now consider the ...
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Is there a “simple” proof of the isoperimetric theorem for squares?

"Simple" means that it doesn't use any integral or multivariable calculus concepts. A friend of mine who's taking a differential calculus course came up with the problem Prove that among all the ...
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759 views

Constrained optimization: equality constraint

I have this very general problem (for $n>2$): $$ \begin{align} & \max Z = f(x_1,\ldots ,x_n) \\[10pt] \text{s.t. } & \sum_{i=1}^{n} x_i = B \\[10pt] & x_i \geq 0 \end{align} $$ Assume ...
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Is this operator monotone?

Consider a convex optimization problem. $$\min_{u\in\Re^k} f(u)$$ s.t. $g_i(u)\leq0,\ i=1,\ldots,m$ Let ...
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287 views

Two Field Problem

Ella Mental has $600$ ft of fencing to enclose two fields. One is to be a rectangle twice as long as it is wide and the other is to be a square. The square field must contain at least $100$ ft ...
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742 views

Why gradient descent works?

On Wikipedia, this is the following description of gradient descent: Gradient descent is based on the observation that if the multivariable function $F(\mathbf{x})$ is defined and differentiable ...
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46 views

Evaluate $f(x_0)+f(y_0)$

Let $$f(x)=3(x-2)^{\frac{2}{3}}-(x-2),~0\leq x\leq 20$$ Let $x_0$ and $y_0$ be the points of the global minima and maxima, respectively, of $f(.)$ in the interval $[0,20]$. Evaluate $f(x_0)+f(y_0)$ ...
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How to minimize cost of group of items given that weights of item sums up to fixed value and atmost 'n' number of items are allowed?

Given that we have a set of items :- { (c1, w1) , (c2, w2), (c3, w3) , ... } where (ci, wi) are the respective cost and weight of the ith item. Its required to minimize total cost of items C such ...
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“Cookbook” methods for neighborhood structure design in simulated annealing for combinatorial optimization?

What are some "cookbook" methods for neighborhood structure design in simulated annealing for combinatorial optimization? Are some reviews or books that contain some "cookbook" methods for ...
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Find $a$ for which figure's area is maximum

Suppose that we have following interval $(-5,2)$,we should find such $a$, which takes all possible values from this interval,creates following inequality systems $$5+a-|2y|\ge 0$$ $$|x|\leq ...
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lagrangian minimisation problem and Karush-Kuhn-Tucker conditions

A rectangular box without a lid is to be made from 50m² of cardboard. Find the maximum volume of such a box.( i know how to solve this in the conventional way, i am trying to figure out how to do it ...
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Maximizing volume of a rectangular solid, given surface area

Maximize the volume of a rectangular solid, given that the sum of the areas of the six faces is $6a^2$ for a constant $a$. So basically they tell you it's a rectangle with 6 sides. 2 sides are ...
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165 views

Find the minimum value of $P=x^2+y^2+\frac{x^2y^2}{(4xy-x-y)^2}$

Given that $\frac{1}{3}<x \le \frac{1}{2}$ and $y\ge1$ Find the minimum value of $P=x^2+y^2+\frac{x^2y^2}{(4xy-x-y)^2}$
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Finding figure of least perimeter

I was stuck (not exactly) at this problem for about a month and finally decided to ask my doubt here. This problem is: Given a family of parallelograms '$R$', all of which are on equal bases ...
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Is concave quadratic + linear a concave function?

Basic question about convexity/concavity: Is the difference of a concave quadratic function of a matrix $X$ given by f(X) and a linear function l(X), a concave function? i.e, is f(X)-l(X) concave? ...
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357 views

Bilinear Optimization Problem

How could I solve the following optimization problem using MATLAB or an other way? Given ${E}^{1}, {A}^{21}, {A}^{22}, {C}^{1}, {A}^{12}$ $ \underset{{C}^{2}, {E}^{2}}{min} {\left \| {C}^{2}{E}^{1} ...
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How to get the initial ellipsoid in the ellipsoid method for solving optimization problem?

If what I assume is correct, assumption : for a maximization problem, we run a binary search over estimated values, starting with max estimated value, and narrow down to the feasible optimal value ...
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Optimization with Tricky First derivative condition

I have a function $f$ such that the first derivative consists of $f'_i=\alpha_i*\beta_i$, where if $\beta_i=0$ then I get linear dependence in my solution (which is not allowed). So for the first ...
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Application of Lagrange Multiplier?

Let $M$ and $m$ denote resp. the max and min values of the func. $f(x,y,z) = xyz $ over the region defined by the interior and boundary of the sphere $x^2 + y^2 + z^2 = 3$. What is the value of $M + ...
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Simple example application of Karush-Kuhn-Tucker conditions to minimization problem

I am wondering if there is a simple example application of the Karush-Kuhn-Tucker conditions to show that a minimum exists for a multivariate minimization/optimization problem. Could anyone suggest a ...
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Question simplex method application criteria?

Can we apply simplex method if one or more equation are equal to zero. tell me full criteria my question example is as follows: Maximize: $z=135x+50y$, subject to: $$\begin{align} ...
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Computing the volume of a set efficiently

Given a set of vectors $\mathbf v_i$ for $i=1,\dots,k$, $\mathbf v_i \in \{0,1\}^n$, is that possible to efficiently find the volume of the set, $$\left\{\mathbf x \in [0,1]^n:\mathbf x \le ...
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What algorithms and/or software libraries should I use to solve this?

I'm trying to write simple data format recognition program (that shows "what things does this unknown uncompressed unencrypted file have inside and where that things are located in the file") and it ...
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What would be the formula to optimize RPS vs. Latency?

I am trying to come up with a cost function that I could minimize/maximize. This would be a two variables function: number of requests per second and latency in ms. What we observed is that the more ...
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Finding an easy way to detemine the minimal distance

Let $c>0$ be a given constant and consider the real function on $\mathbb R^2$ given by $f(x,y)=c(x^2+y^2)$. Is there an easy way (that is, a solution without using Lagrange multipliers) to ...
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Understanding the concept behind the Lagrangian multiplier

I've been trying to understand the principles behind the Lagrangian multipliers and I think I've got a rough understanding of it. Would appreciate it if you guys could help me answer a few questions! ...
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509 views

LP relaxation for ILP\IP (integer linear programming)

I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ...
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Minimize sum of smallest and largest among integers on the real line.

Suppose there are 3 non-negative integers $x$, $y$ and $z$ on the real line. We are told that $x + y + z = 300$. Without loss of generality, assume $x$ to be the smallest integer, and $z$ to be the ...