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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Simple question on maximizing a family of concave functions

Given $x \in [0,1]$, let $f_k(x)$ be concave function in $x$ for all $k= 0,1,...,N$. Is the following two maximization formulations equivalent? $$ \max_{0 \le k \le N} f_k(x) =?= \max \{f_0(x), ...
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Resolve this system:

Im tried to resolve this problem: $$\max\quad f\left( x,y \right) =xy\quad \text{s.a}\quad \begin{cases} x^2 +y^2+z^2 -1=0 \\ x+y+z=0 \end{cases}$$ Well, i form the lagrangian and the respective ...
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91 views

The amazing lift.

I would like to ask for a program to efficiently calculate how a lift should fetch the people who need it. Most of us use lifts (or elevators) but maybe it could be programmed to be faster! Or can it? ...
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Minimum curve for the distance between two points at the plane

The problem is to determine the curve y=y(x) in the plane, the lenght of which is given by the functional: \begin{equation} I(y)=\int_{x_1}^{x_2}\sqrt{1+(y')^2}dx=\int_{x_1}^{x_2}F(x,y,y')dx ...
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Optimization / personalization within clusters

I have the following optimization problem: I have a (random and very noisy) objective function $f(A, P)$, where $A$ is a vector of "observable" parameters of the input and $P$ is the parameters that ...
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84 views

Taylor series approximation of function under norm

I am reading this paper. At page number $4$, term $||Au - f||^{2}$ is approximated by taylor series approximation around $u^{k}$. The resulting approximations are $$\|Au - f\|^{2} \approx ...
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305 views

What numerical methods are known to solve $L_1$ regularized quadratic programming problems?

What numerical methods are suitable to solve the following problem $$\min_x \tfrac{1}{2}x^T A x + b^Tx + \lambda ||x||_1$$ where $x,b\in\mathbf{R}^n$, and $A\in \mathbf{R}^{n\times n}$ is positive ...
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22 views

For given mean $\mu$ of random variable X in [0,1], what is the probability distribution function $p(X)$ that makes $VAR(X)$ maximum?

Given the conditions $\int_{0}^{1} p(x)dx=1$, $\int_{0}^{1} xp(x)dx=\mu$ and $p(x)\ge0$ for $\forall x \in [0,1]$, What probability distribution function $p(x)$ makes $Var(X)$=$\int_{0}^{1} ...
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31 views

How should I modify the initial guess for normalised version of an optimisation problem?

I am solving a large optimisation problem. The problem contains 500 decision variables and few hounderd constraints. Since the constraints are very tight and nonlinear, any random initial guess ...
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144 views

Minimum value of $\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\frac{24}{5\sqrt{5a+5b}}$

Let $a\ge b\ge c\ge 0$ such that $a+b+c=1$ Find the minimum value of $P=\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\dfrac{24}{5\sqrt{5a+5b}}$ I found that the minimum value of $P$ is ...
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Determining the optimal cost through dynamic programming.

There are $n$ houses numbered $\{1, 2, 3, \dots, n\}$. The cost of laying a cable that serves houses $j, j+1, j+2, \dots, j+k $ is $f (j, k)$. One cable can serve a maximum of 10 houses. The ...
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774 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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16 views

reformulating an integer quadratic problem into a linear integer problem

I am trying to solve an optimization problem, in order to find an optimal runtime schedule for a machine. It involves one boolean variable $x_{t} \in \mathbb{\{0,1\}}$, that describes whetever the ...
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40 views

Can I perform Maximum likelihood via optimization?

I have two $3 \times 3$ matrices $\mathbf{a}$ and $\mathbf{f}$. $\mathbf {f}$ is completely known to me. Also $a_{ij} \in [+1,-1]$ \begin{equation} \mathbf{f} = \left( \begin{array}{ccc} f_{11} ...
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A clarification regarding dynamic programming.

This is a question regarding dynamic programming. The document to which I am referring is this (pg 325). It says that $$v_n(s_n)=\text{Min}\{t_n(s_n)+v_{n-1}(s_{n-1})\}$$ Here $v_n(s_n)$ is the ...
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36 views

Iterative method to compute only the positive eigenvalue's and corresponding eignevectors of a very large matrix?

I have a very large dense matrix (~10000 X ~10000) which is not full rank . I want to compute only the positive eigenvalues and corresponding eigenvectors instead of computing all of them. I have ...
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35 views

The Hessian of the maximum of a function

I have an optimization problem where the inequality constraint is in the form of: $\min\limits_x f(x)$ subject to $\max\limits_{\zeta} g(x,\zeta) \leq 0$ where $x$ is the optimization variable. I ...
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22 views

A length of fence encloses an area alongside a river — what is the optimal shape to maximize area? [duplicate]

You have $100$ meters of fence . There is a perfectly straight riverbank, much longer than $100$ meters, so you have plenty of room to work with. What is the optimal shape and dimensions that ...
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11 views

Can a Convex QCQP Problem with an additional linear constraint be converted to a SOCP?

I have a quadratically constrained quadratic programming problem that I massaged into the form $$ \begin{aligned} & \underset{x}{\text{minimize}} & & x^T Q x \\ & \text{subject to} ...
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18 views

Optimization: maximizing nonconvex sum of product of constraints

I'm wondering if there is any way to convexify, approximate, and/or simplify the following problem. $\max. \sum_{k \in K} \prod_{i \in I} (a_{ik} x_{ik} + b_{ik})$ s.t. $x_{ik} \in [0,1]$ where ...
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A repository of constrained optimization test problems?

I am looking for a repository of constrained optimization problems with solutions. I want to find "benchmark" type problems to test my algorithm on and just trying to search for known problems ...
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Product expression maximization given 4 matrices as function of 2 column vectors

Hypothesis: - we are given 4 complex matrices denoted with $H_1, H_2, G_1$ and $G_2$. - the 4 matrices are not necessarily square so their size is $N$ by $M$. - we denote with $w_1$ and $w_2$ two ...
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Nocedal/Wright: Numerical Optimization, Lemma 12.3.(ii)

In the above given monograph (1999, 1E) the following parametrized system of equations $R:\mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^n$ is introduced: $$ R(z,t) := \left[ \begin{array}{c} ...
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Ideal amount of piles to sort a stack of 250 cards (magic the gathering)

I'm a hobbyist working on a mechanical sorting machine to sort magic the gathering cards. I'm by no means a mathematician though, and I was wondering if you all wouldn't mind helping me out with a ...
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31 views

What is topological learning?

I am getting this term topological learning in few places for example a reference is below at section 1.1.2: http://virenjain.org/thesis/VirenJainThesis_official.pdf Can anyone point out what ...
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How to solve the Few Scientists Problem (big word problem) in its general form?

I'm trying to figure out how to solve this word problem. I'm pretty sure it involves calculus or something even harder, but I don't know how to solve the general form. Let me start with the concrete ...
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35 views

Optimisation Problem for Pipe Nesting

I work in a company where we are supposed to produce and send pipes using trucks to buyers. Pipes of smaller diameter can be nested inside pipes of larger diameter while sending to minimize number of ...
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Convex Optimization Problem

Can anyone please solve the following optimization problem? it should be solvable using lagrangian relaxation method.
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Optimization: KKT conditions statement

I'm currently following this material Optimization Theory: Chapter 2 Theory of Constrained Optimization And I can't understand why the following statement is true, between the equations (2.9) and ...
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Given the Hessian of a scalar function of two variables, find the function

Suppose we are given the Hessian of a scalar function of two variables, $\nabla^2_{f(x,y)}$, do there exist explicit formulas for finding $f(x,y)$? Attempt: Brute-Force method is to do double ...
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Choosing a surface that makes the flux of F maximal,

For a closed surface S in $R^3$, consider the flux of F, given by the usual flux integral. For what choice of S will the flux be maximal? So, I want to apply the divergence theorem and instead look ...
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Linear Programming: Three variable graphical solution

A small bank offers three type of loans: housing loans at $8.50$% interest, education loans at $13.75$% interest rates, and loans to senior citizens at $12.25$% interest. Further, it needs to ...
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Module of the differential of a function

Given two triangles, $PQR$ and $P'Q'R'$ in $\mathbb{R}^2$, I want to find a bijection $f$ between $PQR$ and $P'Q'R'$ such that: 1) $f$ maps vertices in vertices and sides in sides (i.e. $P$ in $P'$, ...
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Calculus question with optimization homework

A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (a) How much of the wire should go to the square to maximize the total area ...
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Can a dynamic programming problem be transformed into a linear algebra problem?

Here is a simple standard economic problem: Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
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856 views

Maximize linear equation with 2 variables

How do I maximize the following equation: $$ 150 \le 9.05x + 18.89y \\ \text{constraints: } \\ x > 0, y > 0 \\ \text{$x$ and $y$ must be whole numbers.} $$ I cannot use calculus to solve ...
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Mana Maximization (Hearthstone)

I recently started playing Hearthstone and a statistic / probability question came up my mind. Here's a quick breakdown: The game is a turn-based card game which involves "points" that you can used ...
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How to calculate the hessian of a matrix function

I am estimating a model minimizing the following objective function, $ M(\theta) = (Z'G(\theta))'W(Z'G(\theta))$ $Z$ is an $N \times L$ matrix of data, and $W$ is an $L\times L$ weight matrix, ...
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Quasi concavity and quasi convexity of a max function

Consider the production function: Q = max {K, L}. We have to find out whether this function is quasi concave or quasi convex. According to me, since this function is concave, it should automatically ...
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Solving a matrix equation using numerical optimization

To my knowledge, if $A \in \mathbf{S}^n_{++}$, then given any $b \in \mathbb{R}^n$, the system of linear equations $Ax = b$ has a unique solution $x^* \in \mathbb{R}^n$. Moreover, the solution $x^* ...
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Estimating the “step size” of a grid

Suppose one is given a set of $M$ points distributed on a "grid", i.e: $$x_i = x_0 + \alpha n_i + \epsilon_i, \quad n_i\in\mathbb{Z}$$ This might like something like this: $\quad\ ...
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maximize target function given conditions

Basically the target function need to be maximized has the form $$T = \sum_{i=1}^K d_i* F_i({d_i}, {n_i})$$ with constraint $$\sum_{i=1}^K n_i \le N$$ If $n_i * m_i \lt d_i$ then $F_i({d_i}, {n_i}) ...
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Understanding graphical meaning of tangent line in optimization problem

In a trivial optimization problem where dependent variable $y(x_b)$ is a curve, I'm seeking the value of $x_b$ that minimizes $\frac{y(x_b)}{x_b-x_a}$,where constant $x_a>0$. The solution has been ...
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Ensuring parameters of log-linear model sum 1

I am training a log-linear model with parameters $\theta$ using SGD. I want to ensure that my parameters will end up being probabilities i.e. $\sum_i \theta_i = 1$. One way to do this is by using ...
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Saturation Curve

I have an expression which is $\displaystyle\frac x{x+40}$. I'm trying to find a point indicated in the graph. As you can see I drew 2 lines, one tangent to the region which it saturates, the other ...
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Convex Optimization of quadratic function with inequality constraints

How would I solve the following problem? $$\min_{x\in\mathbb{R}^n} x^T A x$$ subject to the constraints $$x_i\geq 1,\,i=1,\dots,n,$$ where A is positive semidefinite and symmetric. Is it possible to ...
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Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability [duplicate]

Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability $f:\mathbb{R}^n\rightarrow\mathbb{R}$ I don't know how to star this problem, I just know the definition ...
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21 views

Gradient of a forth order scalar function with respect to a Matrix

I'm trying to take the gradient of the following function w.r.t A: $$ f(A) = ||AC_YA^T - C_R||_F^2 $$ I tried the following: $$ f(A) = trace((AC_YA^T - C_R)^T(AC_YA^T - C_R)) = ...
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Simple verification: is this equivalence always true?

I have a constrained optimization problem and I am trying to reduce the number of contraints and am afraid to be losing information by doing so. If we have two constraints as the following $$A \geq B ...
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440 views

Using gradient descent and Newton's method combined

I have this function $f(\mathrm{X})$ where $\mathrm{X=A+B+C}$ where $\mathrm{A}$ is a diagonal element with variable $a$ on its diagonal. $\mathrm{B}$ is another diagonal matrix with variable $b$ on ...