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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Optimize distributions for low mean, high variance

Assume a context with $N$ approximately normal distributions where a lower mean implies a 'better' distribution and a high variance or high standard deviation implies a 'better' distribution as well. ...
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How to resolve total variation $F(u)=f(u)+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$

Given $u(x)=[u_1(x)..u_N(x)]$, $0 \le u_i(x) \le 1, \sum_i^N u_i(x)=1$ and the cost function is: $$F(u)=f(u(x))+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$$ where $u_i(x)$ is a value that indicate ...
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Optimization problem for maximum volume of inscribed figure. [duplicate]

While studying, I came upon this problem: "What is the largest possible volume a right circular cylinder can have if it is inscribed in a sphere of radius 5?" The answer was shown as ...
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Numerically/Computationally estimating parameters

I have a function $f(x)$ and I have an estimating function $\hat f(a,b,c,d;x)$ Say, I also have a scoring function $S(f,\hat f,x)$ (which could very well be mean square error) And I have some ...
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optimization of a function with inequality constraint

I have a function to be maximized subject to constraints. I can write the primal Lagrange function as the following: (objective function WITH two constraints in the last two terms) $$L_P = ...
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Is there any method that convert a concave problem into convex problem?

I have an optimization problem of the form: \begin{align} \begin{cases} x_2 \rightarrow \min, \\ \text{subject to:} \\ f_1(x) \leq 0, \\ f_2(x) \leq 0, \end{cases} \end{align} with $x= (x_1,x_2)^T$ ...
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optimizing over a set of symmetric matrices

I need to minimize a complicated loss function, $f\left(\Lambda\right)$ over a set of symmetric matrices, $S_{p}$ of dimension p, such that all the eigenvalues of $\Lambda \in \left[0,1\right]$. I ...
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295 views

optimization of a non-differentiable, component-wise step function

I would like to estimate the (local) minimum of a function $c:R^N \mapsto R^+$ where: $c$ is only differentiable almost everywhere, there exists a component $j$, such that $\frac{\partial ...
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427 views

How to see that K-means objective is convex?

I'm trying to proof that the objective of the K-means clustering algorithm is non-convex. The objective is given as $J(U,Z) = \|X-UZ\|_F^2$, with $X \in\mathbb{R}^{m\times n}, U\in ...
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Reliability/survival function raised to a power

Let $r(p)$ be the reliability function, and suppose that $r(p)=r(p,p,...,p)$ and that $r(p_0)=p_0$ for a certain $p_0$, $0\leq p,p_0\leq 1$. I'm asked to prove that $r(p)\geq p$ if $p\geq p_0$ and ...
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multi-objective reduction of a given set

I have a set of arguments $v_k$. Each argument has a set of two different numeric values $x_{ak} \in [0,\infty]$ and $x_{bk} \in [0,\infty]$ associated to it. The set $V$ contains all $v_k$s. I’m now ...
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305 views

Baseball Roster Optimization

I'm trying to programmatically optimize a Fantasy Baseball Roster that requires a fixed number of players at position (2 Catchers, 5 Outfielders, etc.) and has a salary constraint (total draft price ...
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52 views

Campbell's Source coding

In the usual Shannon's source coding problem one chooses code words that minimize $E[L]:=\sum_i p_il_i$ over all $L=(l_1,l_2, \dots), l_i\ge 0$ such that $\sum_i e^{-l_i}\le 1$ (Kraft inequality), ...
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How to optimize the repartition of samples in weighted channels?

This is more like an applied mathematics question, so my apologies if I am at the wrong place. Let S(n) be an infinite sequence of real numbers strictly growing from 0 to 1 (asymptotically). Let P be ...
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Is this function convex or non-convex? How do you decide?

The problem is: find $$\min⁡ \mathrm{P}\left[{\log(1+p||H^H \mathbf{w}||^2)\over 1+p||G^H \mathbf{w}||^2}<R\right]$$ constraint to: $||\mathbf{w}||^2=1$ where $H$ and $G$ are matrices of ...
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Optimization problems with combinations of a finite set as the feasible area?

For example: Provided that $S\subset \Re$ is a known finite set ($n\leq |S| < \infty$), number $k$ is known, and $1 \leq k<n$ minimize $f(x_{1},\ldots, x_{n}) = \sin (\sum_{1\leq i\leq ...
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projection KKT optimal condition

Using the KKT optimality condition find the orthogonal projection of an arbitrary point $c \in$ to the closed convex set $C$ (non empty) defined by: (a) $C=\{x \in R^n : Ax\leq a\}$ where $A\in ...
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36 views

Local extrema in special directions

I am looking for the extrema of a function $G(y_1,y_2,y_3,y_4)$ subject to the constraint $y_1 = y_4 + y_2y_3.$ We know that $G$ is defined if $(y_2,y_3,y_4)$ is in the cylinder $\mathbb{D} \times ...
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find the minimum value of this integral when $1>t>0$, $f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x = ?$

Is there someone who can show me How do i find the minimum value of this integral when $1>t>0$, \begin{align*}f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x &= \end{align*} Note : ...
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KKT Optimality Conditions

I am working with the following optimization problem: $$ \min_{\Delta} \boldsymbol{\theta}^T\boldsymbol{\Delta} \\ \text{Such that:} ~~~0 \leq \mu_i + \Delta_i \leq 1 ~~\forall~~ i\in\{1,2,\ldots, n\} ...
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453 views

Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
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435 views

Can there be a cubical bubble?

Although not physically perfect, a reasonable mathematical model for a bubble's shape is that it minimizes surface area subject to fixed volume. A single floating bubble is usually a sphere, but ...
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45 views

Generating vectors in a non-orthogonal 3D lattice with increasing magnitude

I am trying to build an algorithm to generate a sequence of lattice vectors $\mathbf{v}_n$ in 3D such that: (a) the first vector $|\mathbf{v}_1|$ is the shortest vector of the lattice (b) for all $i ...
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E-olymp: Cake. Giving Wrong Answer

Cake This is a e-olymp programming question mathematical optimization. In honor of the birth of an heir Tutti royal chef has prepared a huge cake, that was put on the table for Three Fat Man. ...
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A special case of GUBMKP

I am seeking a problem that resembles the Multidimensional Knapsack Problem with Generalized Upper Bound Constraints where the resources available are of equal sizes.I am only getting the case where ...
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Optimization of solve

Find the minimum value and the maximum value of the function $$y(x)=\frac{x^3}{x-3}$$ when $4\le x\le5$ I found that $f(x)$ is decreasing on the interval $[4,\frac{9}{2}]$ and increasing on ...
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Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
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Do we really need the constraint qualification?

I can't keep my fingers off Nocedal/Wright's Numerical Optimization (1999,1E) and I apologize. But maybe YOU can shed light on the question: Why does a point $x \in \mathbb{R}^n$ need to satisfy the ...
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Minimizing Area by Approximation

Suppose I have an increasing step function $E_c$ given by $$E_c(\phi) = \sum_{i=1}^n E_i \theta(\phi - \phi_i),$$ where $\theta$ is the Heaviside step function and $E_i$, $\phi$, and $\phi_i$ are all ...
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Coin distribution problem to optimize

There are $N$ users, with each user having a money request. There are $T$ coins, these coins are to be assigned to the user in such a way that its request is fulfilled. Assume each coin may have ...
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Is it covex function?$J_{new}(u)=\int_{\Omega} \sum_{i=1}^{N} \lambda_if(x)u_i(x)dx$

I have a function such as $$J(u)=\int_{\Omega} \sum_{i=1}^{N} f(x)u_i(x)dx$$ where $f(x):\Omega \to R$, $0 \le u_i(x) \le 1,\sum_i u_i(x)=1$ Given that $J(u)$ is a convex function w.r.t $u$. Now I ...
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How does one evaluate the derivative of a matrix with a tensor $\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}$?

I am stuck on the following: $$\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}=\text{ ?}$$ with $A$ a $d\times d^2$ matrix, $\mathrm{Id}$ the identity matrix of $d\times d$ ...
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Minimizing the function with a log determinant and trace function?

I am trying to minimize the following argument, which is unbounded in case one of the eigenvalues of $A$ is equal to zero. $\arg min_{S} \log|S^H A S| - tr\{ \Sigma^{-1}S^HAS\}$ Let $A > 0$, ...
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optimal control, semismooth newton, bounded norm

I'm solving an optimal control problem (Poisson's equation with dirichlet BVP) $F(y,u) :=\frac{1}{2}\int_{\Omega} (y-y_d)^2 dx + \frac{\lambda}{2} \int_{\Omega} u^2 dx$ with finite element method. ...
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Linear independence of equality constraint gradients in constraint qualifications

I'm, trying to get an intuitive feel for the various constraint qualifications for KKT points. Most of them seem to rely on the linear independence of $\nabla g_i(x^*)$ where $g_i$ are the equality ...
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Usefulness of prime numbers as Threading Timeouts in programming [closed]

I am a .NET programmer, founded in math. I am having an argument with a fellow programmer. When I add a Threaded Timer to the program, the interval in milliseconds I use is always a prime number. ...
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Convergence results for block coordinate descent methods

I am trying to solve the problem minimize $f(x)$ subject to $x_1 \in C_1, x_2\in C_2, ... x_m\in C_m$ where $x_1, ..., x_m$ are block subvectors of $x$, and $C_i$ are each closed convex sets (not ...
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what are Smooth and Non Smooth Problem in optimization?

I am trying to understand the difference between the optimization problem types which are basically smooth and non smooth. I also found this question what does a smooth curve mean? I understand that ...
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Maximization: KKT on unbounded region

Solve the following NLP: $$\left\{\begin{matrix} \min & -3x+y-z^2\\ s.t& g(x,y,z)=x+y+z \leq 0\\ & h(x,y,z)=-x+2y+z^2z=0 \end{matrix}\right.$$ My attempt Using kkt conditions, we ...
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541 views

Efficient Cholesky decomposition of inverse matrix

I want to generate random numbers from a multivariate normal distribution in Matlab. Normally, this is done like: $w = \overline{w} + \text{chol}(\Sigma) \cdot \vec{l}$ But in my case I don't know ...
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Optimal approximation of quadratic form

Let $\mathbf{x}\in\Bbb{R}^n$ and $A\in\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ denotes the space of symmetric positive definite $n\times n$ real matrices. Also, let $Q\colon\Bbb{R}^n\to\Bbb{R}_{+}$ be ...
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non-linearity and non-convexity

I am taking a course on linear regression online and it talks about the sum of square difference cost function and one of the points it makes is that the cost function is always convex i.e. it has ...
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Matrix norm in the objective of an optimization problem

I am stuck with the following optimization problem from research. The optimization problem have the following objective function: $\|Q-H\|_\infty$. Here $Q$ is a PSD matrix and $H$ is a symmetric ...
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multi objective optimization

Suppose we want to maximize two positive bounded objectives. A usual approach for this aim is to maximize a weighted sum of these two objectives. Now, my question is why not to maximize the product ...
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Eliminate cases before calculting all KKT conditions

I have the following non linear programming to solve: $$\left\{\begin{matrix} \min & (x-3)^2 + (y-2)^2 \\ s.t. & x^2 +y^2 \leq 5 \\ & x+y\leq 3 \\ & x \geq 0\\ & y\geq 0 ...
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An exam`s points dilemma

On July 2 I have an exam, in this exam will be 40 questions in test with 5 variants of answer for each question. For each correct answer will be given +1 point. For each incorrect answer will be ...
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What is the 'optimal' equal-area partition of a circle?

What is the (an?) n-partition of a circle that meets the following criteria: The boundaries of each partition can be represented as a union of finitely many finite-piecewise-smooth simple closed ...
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Find $\alpha$ which makes the problem have an optimal solution or none at all + dual problem LP

min $x_1 + \alpha x_2 $ subject to $4x_1+3x_2\leq29$ $x_1+x_2\geq4$ $x_1\leq5$ $x_2\leq7$ Find for which $\alpha$ the given problem has an optimal solution or no solution at all. Provide the ...
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Estimate Beam and Ball Problem System Parameters

I'm trying to estimate the parameters of beam and ball problem model. In the problem we have output as ball position and input as gear rotation angle. The issue that i want to ask is that our ...
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If the iteration $x^{k+1}=x^k-t_kH_k^{-1}\nabla f(x^k)$ converges superlinearly to a stationary point $x^*\ne x^k$, then $t_k\to 1$

Let $f\in C^2(\mathbb{R}^n)$ $(H_k)_{k\in\mathbb{N}_0}\subseteq\text{GL}_n(\mathbb{R})$ $x^0\in\mathbb{R}^n$ and $$x^{k+1}:=x^k+t_k d^k\;\;\;\text{for }k\in\mathbb{N}_0\tag{1}$$ with ...