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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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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24 views

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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55 views

Solving $ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$

Let $\phi \in \mathcal ( [0,1]^2)$ symetric , can we find a solution to the following minimisation problem? $$ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$$ with $$ ...
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20 views

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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40 views

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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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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61 views

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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54 views

Half Sphere Optimization

Having a little trouble with an optimization question: ...
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4answers
81 views

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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153 views

$|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2}$

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 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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27 views

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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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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44 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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1answer
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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1answer
26 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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23 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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3answers
57 views

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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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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29 views

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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55 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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28 views

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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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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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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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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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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57 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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32 views

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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58 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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36 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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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
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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54 views

$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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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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3answers
117 views

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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135 views

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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Maximize are of rectangle with semicircles on left and right [closed]

There is a rectangle with semicircles on the left and right sides. You know that the perimeter is 100. Maximize the area of the entire shape.
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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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40 views

find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.