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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Hessian of non-differentiable function

Given a function $f = \max\{f_1,f_2\}$ with $f_1,f_2$ convex and differentiable, I know I can calculate the subgradient of $f$. Is there also an equivalent of the subgradient for the (sub)Hessian? ...
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Convergence of projected gradient method for non-convex functions.

Is there a proof of convergence for the projected gradient method for non-convex functions? By projected gradient method I mean the following (shortened) algorithm for $f: U \rightarrow \mathbb{R}$ ...
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Restating optimization problem for quadratic programming

I'm working on implementing an author disambiguation algorithm as described in Torvik et al's paper. I've got most steps done, but am completely stumped on implementing a quadratic optimization step. ...
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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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Optimization of parallelepiped.

Let $K \in R^3$ the ellipsoid given by the equation $ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $ with $a,b,c > 0$ , let $(x,y,z) \in K$ on the first octant, consider the ...
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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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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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Finding $p$ of the binomial cdf…

Please bear with me, I'm only a biologist ^.^: I have a need of solving this cdf so as I can plug in known values $Pr, n, k$, and get an answer for $p$. $$f(k;n,p) = Pr(X\le k) = \sum_{i = ...
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22 views

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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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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55 views

Half Sphere Optimization

Having a little trouble with an optimization question: ...
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261 views

Minimum number of lines covering n points

Let there be n points in the plane. I want to know the minimum number of horizontal and vertical lines covering all the points in the plane. My initial approach started like this, 1) for each point I ...
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Matlab Optimization problem with Matrices

I'm trying to solve an optimization problem in Matlab. The equations 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 x 1) mu_BL (1 ...
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Proof that feature normalization cause faster convergence of gradient descent

How to prove that if I do feature normalization (scaling of the $x_1,\ldots,x_n$ to be all in range $[0,1]$) to a convex function $f(x_1,\ldots,x_n)$ that returns real scalar, then gradient descent ...
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48 views

Find the angle between hypotenuse and the side.

For a right angled triangle, the sum of the length of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle. Find the angle between hypotenuse and the side.
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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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45 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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139 views

formulate this scheduling problem as linear programming problem

Sorry if this very silly, but i am something new to optimization theory: We have $m$ identical Machines and $n$ jobs. A job $j$ can be done in any of the identical machines in $p_{j}$ time units. ...
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30 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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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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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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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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A primitive function of $ e^{x^{2}} $

I made some efforts to set a closed form of primitive function of $ e^{x^{2}} $ i find this function : $ f(x)=\frac{x}{2x^{2}-1}e^{x^{2}} $ where : $f'(x)=(\frac{x}{2x^{2}-1}e^{x^{2}})'$= ...
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dual form of an optimization

Consider the following optimization in primal form $\displaystyle\max_{x_1, \ldots, x_n}\sum_{i=1}^n d_ix_i -\sum_{i=1}^n x_i\log(x_i)$ subject to $a_i\leq x_i\leq b_i$ and $\sum_{i=1}^n ...
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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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68 views

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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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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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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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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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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Convergence of Gradient method

Is it guaranteed that 'gradient type' methods (like Ellipsoid method) converge to KKT point? Actually I am trying to optimize a non-convex problem with ellipsoid method, I know that KKT satisfaction ...
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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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How to find the minimum value of $x^2+y^2+xy-4$ where $x+y=2$. [closed]

How to find the minimum value of the expression: $x^2+y^2+xy-4$ where $x+y=2$
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Global consistency of constraints in a MIP program

How does a Mixed Integer Programming (MIP) solver ensure global consistency of constraints while adding an additional constraint (during branch and bound). A naive method would be to add 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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42 views

Find max and min on some region

Find maximum and minimum values of $f(x,y,z)=x^2yz$ on the region $x^2+y^2\leq1,$ $0\leq z\leq1.$ First, I get $\nabla f= (2xyz, x^{2}z, x^{2}y) = (0,0,0) \implies x = y = z = 0$, so the critical ...
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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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27 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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21 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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Finding extremes on set with one constraint

I have $f(x,y)=x*y*e^{-x^2-y^2}$ and I have set $A=\{[x,y]\in \mathbb{R}^2,x^2+2y^2\ge2\}$. I have to find extremas on set A. How do I do it? It is first time when I am encountering problem with only ...
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114 views

Approximate Periodic Function by shifting Basis Functions

Given a periodic "Target Function" $F(t)$ a set of $N$ periodic "Basis Functions" $B_i(t)$ of arbitrary shape All functions are defined on the same interval $T$. I am allowed to shift ...
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Lagrange Method Problem

I am from engineering background and I am currently studying calculus. I had a question from assignment to be solved from a course on coursera but I could not do it. People have posted solution in the ...
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86 views

Real estate problem - local maxima

A real estate office manages $50$ apartments in a downtown building. when the rent is $\$900 $ per month, all units are occupied. for every $\$25 $ increase in rent, one unit becomes vacant. on ...
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The sum of two variable positive numbers is $200$. Find the maximum value of their product.

The sum of two variable positive numbers is $200$. Let $x$ be one of the numbers and let the product of these two numbers be $y$. Find the maximum value of $y$. NB: I'm currently on the ...