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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maximum and minimum values of a function

HI! I am currently working on some calc3 online homework problems and this one is giving me a bit of tough time. I found the gradient of f to be <16x,10y> and the gradient of g to be <4,20>. I ...
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Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
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Extremum of a function under constraints

I have a function $f : E \subset R^n \to R$. $E$ is compact and $f$ is continuous so the extremums exist. But $E$ is not defined by an equation but an inequation, so i can't use the Lagrange method ...
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maximizing a quadratic over linear function

Recently I am trying to solve the following optimization problem: $$ \begin{array}{cl} \text{maximize} & \frac{\left(c_1^T x\right)\left(c_2^T x\right)}{d^T x}\\ \text{subject to} & a^Tx\leq b ...
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Half Sphere Optimization

Having a little trouble with an optimization question: ...
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Local minimum of $f(x) = 4x + \frac{9\pi^2}{x} + \sin x$

What's the minimum value of the function $$f(x) = 4x + \frac{9\pi^2}{x} + \sin x$$ for $0 < x < +\infty$? The answer should be $12\pi - 1$, but I get stuck with the expression involving both ...
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$|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2}$

Let $f\in C^1([0,\pi],\mathbb R)$ such that $\displaystyle\int_0^\pi f(t) dt=0$ Prove that $\forall x\in [0,\pi],\displaystyle|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2(t)dt}$ Failed ...
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Matlab Optimization problem with Matrices

I'm trying to solve an optimization problem in Matlab. The expressions you will find below. Problem is it is all matrices, and I have no idea which solver to use for that. $w$ is of size $n \times 1$, ...
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613 views

finding points with maximum distance between them on a circle

I'm a computer science student working on a problem in computer graphics and looking for a formula that can find the x and y positions of a set of N points on the surface of a circle so that the ...
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87 views

Reduce problem to max flow

I have the following question: Assume each student can borrow at most 10 books from the library, and the library has three copies of each title in its inventory. Each student submits a list of ...
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1answer
52 views

Maximum payoff for safe bet

I'm having a hard time choosing a good strategy for this problem: assume that you have $m$ money that you can bet on $n$ mutually exclusive outcomes, all with unknown probabilities, and that each ...
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73 views

How to solve Bellman's optimal equation from the first principle

How to solve the following set (finite) of equations $$ v_*(s) = \max_{a\in A(s)} \sum_{s'} p(s'|s,a) [r(s,a,s') + \gamma v_*(s')]$$ $p$ and $r$ functions are given.
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Extrema of $f(x)=\frac{\sin (5x)} 5 - \frac{2\sin(3x)} {3} + \sin (x)$.

(a) I need help in finding maxima and minima of the following funcion: $$f(x)=\frac{\sin (5x)} 5 - \frac{2\sin(3x)} {3} + \sin (x)$$ therefore I need to find the roots of ...
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Deriving stationary points using the second order derivative.

Suppose that for some function $f$ we want to know the stationary points, i.e. $\frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} = \mathbf{0}$. We can define a new function ...
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322 views

Is Minimax equals to Maximin?

Consider a loss funcation $\ell(x,y)$ with a penalty $g(x,y)$ If I want to consider the worst case robust scenario, that is \begin{equation} \min_x \max_y \ell(x,y) + g(x,y) \end{equation} Is it ...
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How to find out the closed form of a function from its parametric form?

In general suppose that we have a parametric curve given by: $$ x = \phi(t) \\ y = \psi(t) $$ Then if $\phi^{-1}$ exists it is easy to get $y$ as a function of $x$ in closed form: $$ y = ...
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Dual norm equivalence?

$\|\|$ is a norm in $R^n$, its dual norm is defined as $\|s\|^*=max_{\|x\|=1}s^Tx$. We denote $s^\#$ as any vector in the following set: [Arg $max_x: \ \ s^Tx-\frac{1}{2}\|x\|^2$] How to verify ...
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About a minimum-norm problem.

I am studying on the optimization via vector method. The reference book is Optimization by Vector Method by Luenberg. I have trouble in understanding the following statement [p.123]; We consider the ...
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Local minimum implies local convexity?

Consider a real function $f$, and suppose it has a local minimum at $a\in \mathbb R$. It typically looks like What hypotheses can be added to $f$ so that there is some $\epsilon >0$ such ...
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function induced by optimization

Consider the following optimization $\displaystyle\max_{x_1, \ldots, x_n}\sum_{i=1}^n x_i y_i -\sum_{i=1}^n x_i\log(x_i)$ subject to $a_i\leq x_i\leq b_i$ and $\sum_{i=1}^n x_i =c$ ...
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How to find a line that minimizes the average squared perpendicular distance from the given points to the line?

I have set of points scattered around the origin. How to find a vector, such that the average squared distance (perpendicular distance) from points to the vector is minimised? Added For example, ...
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Minimizing $f(x,y,z)=\dfrac{|x|+|y|+|z|}{xyz}$ on a sphere

I need to find the minimum of the function: $$f(x,y,z)=\dfrac{|x|+|y|+|z|}{xyz}$$ with the condition: $$x^2+y^2+z^2=r^2$$ Using numerical methods it's quite easy to solve the problem. How can I ...
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523 views

Optimization problem, not sure how to proceed

So I'm a bit confused by this optimization word problem. I would be able to solve it I think given number values for the speeds but I'm uncertain how to get an exact answer when you don't know the ...
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Help getting a closed-form solution to a maximisation problem

I'm working through a maximisation problem that I can't seem to get a closed-form solution to. It may be the case that there is no closed-form solution, but I would like a second opinion, since I've ...
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Optimal VCV matrix solution of multivariate loglikelhood

I asked a related question yesterday and got a brilliant answer from Ross B. However I still have difficulties. I have the following analog of multivariate loglikehood function (minus 2*log-likehood ...
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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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39 views

show that M isn't close map

the line search map $M:En\times E_n \rightarrow E_n$ defined below is frequently encountered in nonlinear programming algorithm.the vector $y∈ M(x,d)$ if it solves the following problem where $f:E_n ...
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476 views

How to find what are the points closest to and farthest from (0,0) of ellipse $9x^2+4y^2=36$ using optimization?

Please do not use Lagrange multipliers. Assume these have not been introduced and optimize. Edit: I try optimizing the squared distance formula using the equation as a constraint, but I only get one ...
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1answer
210 views

How to solve the dual problem of SVM

By solving the primal form of SVM (support vector machine), we can get the dual form of this problem. The more details are shown in wiki of SVM. Given this dual problem, how can I solve the ...
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The Dual problem of a non constraints problem?

The primal problem is $min_{w\in R^d}: P(w)$ where $P(w)=\frac{1}{n}\sum_{i=1}^n\phi_i(w^Tx_i)+\frac{\lambda}{2}||w||^2$. The dual problem is $max_{\alpha\in R^n}: D(\alpha)$ where ...
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$y(x) = \int_0^x \frac{\sin(t)}{t}dt $

Let $y(x) = \int_0^x \frac{\sin(t)}{t}dt $ find maximums and minimums of $y(x)$. First let $F(x) = \int_0^x \frac{\sin(t)}{t}dt$ and $f(t) = \frac{\sin(t)}{t}$ then $F'(c) = f(c) $ then if $ ...
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Finding minimum of a distance function using matlab

I have a function for that I want to find the minimum. The function calculates the distance between two sets where a set is defined as matix of row vectors $ D = [ d_1, d_2, ..., d_n]$, $d_n$ is a $m ...
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116 views

prove that a black-box multivariable problem is convex or concave

First of all I am not mathematician. I want to solve a very complex black box function with several constraints having to do with electrical power flows in electrical grids. At the moment I use the ...
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In constrained optimization problems, when is 'naive' substitution possible?

To motivate the question, consider the following constrained optimization problem: $$ (P1)\quad \underset{(x,y)}{\min} f(x,y)=x^2 +y^2 \ s.t.\ (x-1)^3 = y^2$$ By replacing the constraint $y^2 = ...
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Constraint to unconstraint optimization problem by subsitution

Given the following convex optimization problem $\min_{x,p} ||x|| - p$ subject to $p > 0$ Can I change the above to an unconstrained convex optimization problem by substituting $c = ...
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Clarification on optimization problem

While reading a combinatorics paper about packing densities in compositions, I encountered the following optimization problem. Maximize $$f(\alpha_1, \dots, \alpha_5) = \sum_{1 \le i < j ...
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Steiner tree problem in 3D?

Steiner tree problem in the plane (2D) is explained on wiki that though there's no straight solution, the solution has some properties, namely points added to the graph (Steiner points) must have a ...
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Maximum likelihood estimator transformed parameter

I don't get the gist of b). What is it that we are in fact calculating here? I don't get why we can just plug in the rearranged formula.
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find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$ [closed]

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
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Method of Lagrange multipliers to find all critical points of a function

I am having difficulties in understanding the steps/method required to find the critical points of a function using the method of Lagrange multipliers. I have read through my text book and tried my ...
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1answer
39 views

Could I get the explicit solution to the following problem relate to generalized rayleigh quotient?

$\bf x$ and $\bf a$ are complex vectors, $\bf C$ is positive definite complex matrix, $\bf B$ is positive-semidefinite complex matrix. What's the objective value? Thanks! $$\max_{\bf x} ...
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Does convex optimization belong to linear or nonlinear programming?

Does convex optimization belong to linear programming or nonlinear programming? Is convex optimization an undergraduate topic or a graduate topic?
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extrema of funcion

$f(x,y,z)=x+2z$ and $M=\{[x,y,z]\in\mathbb{R}^3:x^2+2y^2=4,z+y\le 1\}$. I found out that M is not bounded from below so it does not have minimum or infimum. But how do I find maximum? I tried to use ...
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Calculate minimum perimeter of a rectangle with an extra constraint.

I have been set this problem, and although I can derive a minimum perimeter using calculus, I now need to add an extra constraint to one side of the rectangle and I am having problems deriving a ...
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1answer
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Minimize Energy using Gauss-Seidel method with successive over- relaxation.

I have an energy function to minimize $$E = \sum_i \|I_i - \mathbf N_i^T\mathbf L\|^2 + \lambda\sum_{i,j}\|\mathbf N_i - \mathbf N_j\|^2$$ where $I$ is a known scalar, $N$ is an unknown $n\times3$ ...
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Find extrema of $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$

Let $\overline{B_1(0)}\subseteq\mathbb R^3$ be the closed unit-sphere and $a\in\mathbb R^3$. Find all extrema of the function $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$ depending on $a$. ...
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85 views

Finding max/min through lagrangian

I am trying to solve this problem, but I am doing something wrong: $$f(x,y,z)=x^2-y^2,M=\{[x,y,z]\in\mathbb{R}^3:x^2+y^2+z^2=9,x+z\ge1\}$$ And let $g(x,y,z)=x^2+y^2+z^2-9$. Set M is closed and ...
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Maximize the expected values of a function with constrain

Consider $p_1,p_2,...p_N$ are probabilities arranged in ascending order. $n_1, n_2,...n_N$ are numbers which are arranged in geometric progression. I want to Maximize E= $\sum\limits_{i=1}^N p_i\cdot ...
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60 views

Calculus optimization quick question

A hotel fills only $120$ rooms when the price is $\$150$ per night for a room. When the price is decreased by $\$10$, it fills $16$ additional rooms. Find the price for maximum revenue. Ok frankly ...
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49 views

connectivity on graph

Given a graph, undirected or directed, what is the optimal or just good algorithm for finding the following? 1) Whether two vertices are connected. 2) The shortest path going from one to the other. ...