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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The Jeep Problem with Equally Spaced Stations

Consider the following problem. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is required to cross a ...
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From constrained to unconstrained optimization

I have the following convex optimization problem: \begin{equation}\label{prob} \begin{aligned} &\underset{{\bf W, \xi}}{\text{min}} & \frac{1}{2} ||{\bf W}||_2^2 + \sum_{i=1}^n C_{y_i}\max(0,...
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Formulation of constraints

I would like to formulate the following constraints in a tractable form so that I can perform an optimization over the decision variables $A,x_i,y_i$: $$ A + \sum_{i=1}^N x_i D_i + \beta \big(\sum_{i=...
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Developed a function optimization strategy - need opinions

I've developed a function optimization strategy which is close to evolutionary optimization strategies. It works fine for various functions, but cannot be used with thorough success for functions with ...
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34 views

The Jeep Problem and Nash's Friends

The classical jeep problem is the following. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is ...
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28 views

Existence of absolute maxima and minima

In which of the following functions can be guaranteed the existence of absolute maxima and minima? a) $f(x,y,z)=x+y$ with $z\geq x^2+y^2+1$. b) $f(x,y)=\ln (x^2+y^2+1)$, with $x\geq 0$ and $y\geq 0$...
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What is the coordinate of the maximum value of a quadratic function given by two points and axis?

There are only three pieces of information available: the graph passes through (0,0) and (6,0) the symmetry axis is $x$ = 3 the graph is downward My attempt: I've tried to work on ...
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Pythagorean Theorem to Optimize Multiple Variables?

I'm not sure if this is an already established thing or something I just made up that feels good. I have a list of board games that I'm interested in buying based on their price, their overall ranking ...
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Big Balloon Game

The problem In this game, you are given empty balloons one by one, and for each balloon you are to inflate it with air until you are satisfied. If it does not burst, you gain happiness points ...
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12 views

Maximum Likelihood Estimation: Multivariate Gaussian function. Matrix calculus

I am reading a paper and trying to understand how the authors estimated the standard errors of a set of parameter estimates $[\delta\ \ \phi \ \ \Sigma]$. Below is the loglikelihood function (sorry I ...
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3answers
71 views

Maximizing $f(0)$ given that $f(3)=5$ and $f'(x)\ge1$ [on hold]

Let there be $$f:(-1,4)→ R$$ $$\text{differentiable on} (-1,4) , f(3)=5 , f'(x)≥-1$$ $$\text{which is the maximum value of}$$$$f(0)$$
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Subset of Coins with maximal value

Let $ n \in \mathbb{N} $ with $ n\ge 3 $ be given. Assume that you have $ k-1 $ coins of value $ 1/k $ for all $ k \in \lbrace 2,\ldots,n \rbrace $. Now you have to choose a subset of these given ...
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1answer
33 views

Determine the absolute and local extreme values to two decimal places for $y = x^3 + 2x^2 - x + 6$

After finding the first derivative which is $y'=3x^2 + 4x -1$, I found the two $x$ values after using the quadratic formula to factor the above when $y'=0$ which were $x = 0.22$ and $x = -1.55$. From ...
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38 views

Maximising sum of sine/cosine functions

I have got a problem and I would appreciate if one could help. I have to maximise following function that is the sum of sine/cosine functions: $$ f(x,y)=a_1 \cos(x) +b_1 \sin(x)+ a_2 \cos(y) +b_2 \...
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1answer
14 views

How to minimize $(p_1^2 + (1-p_1)^2)^n$ where $p_1 = 1-(1-(k/n))^N$

Consider $S_{n,N,k} = (p_1^2 + (1-p_1)^2)^n$ where $p_1 = 1-(1-(k/n))^N$. If we fix $N$ and $n$, how do we find a $k$ which minimizes $S_{n,N,k}$? We assume that $1 \leq k < N$ if that makes a ...
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1answer
20 views

Gradient of a maximum

How do you compute the gradient of a function that involves a maximum? For example, I have the function: $$ f(\vec{t}) = v(1-\exp(-\lambda\cdot \max(t_0,t_1)))$$ With $v$ and $\lambda$ constant, for ...
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How to solve an inverse problem $d=Ax_1 + Ax_2$

In the optimization problems, there is an operator, $A$, which transforms the model, $x$, to the data domain, $d$. Generally, we don't know the model and we are trying to find it according to the ...
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Approximation of non-differentiable optimization problems with max function

The book by D. Bertsekas "Constrained optimization and Lagrange multiplier methods", Ch. 5.1.3 describes at p. 312 a method that is used to solve non-differentiable optimization problems by ...
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3answers
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How can I solve an optimization problem $x^T A x$ with constraint $x^T x = 1$?

Let $A \in \mathbb{R}^{n \times n}$ be a positive definite matrix. \begin{align} &\operatorname*{minimize}_{x \in \mathbb{R}^n} & & x^T A x \\ &\text{subject to} ...
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Solving algebraic Riccati Like equation using Newtons method

I am trying to solve the following equation for $P$ $0=X(t)^{\rm T}\left(Y^{\rm T}P+PY-\gamma P-PZR^{-1}Z^{\rm T}P+S^{\rm T}QS\right)X(t)+\mu^{\rm T}R\mu,$ where $Y\in\mathbb{R}^{n\times n}$, $Z\in\...
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15 views

Projected gradient descent with momentum

Can we apply momentum to projected gradient descent? If so, how should we do that? In the domain I'm working on, momentum greatly speeds up gradient descent. However, I want to do projected ...
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1answer
35 views

Shortest possible distance to locate an unknown road

You are stranded in the middle of a large desert and the only way home is a through a straight road, which unfortunately you do not know the location of. If the perpendicular distance from you to ...
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Generate a class of matrices via optimization

I want to generate a matrix (using Matlab) with the following properties: (1) $A = (a_{ij}) \in \mathbb{R}^{n \times n}$; (2) $a_{ij} \in \{0,1\}$ and $a_{ii} = 0$ for all $i\in\{1,2,\cdots, n\}$; (...
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Find point on a line that is nearest to the origin

Can you help me with this exercise? Find the nearest point to the origin $(0,0,0)$ in the line given by the intersection of planes $x+y+z=2$ and $12x+3y+3z=12$. The intersection of the planes is ...
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1answer
69 views

Upper bound of $x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2$

Let $n\geq 3$ be a positive integer and let $x_i$'s be non-negative real numbers with $x_1+x_2+\dots+x_n=1$. What is the maximum of $x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2$? If the sum were symmetric ...
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Relationship between Newton's method in root finding and optimization

In both root finding and optimization, there are Newton's method. Wikipedia has 2 links here and here. Root finding is using first order derivative and optimization is using Hessian. What's the ...
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How to cover a sphere with caps removed, with equidistant points?

I have a sphere with the caps removed, so: $$x^2 + y^2 + z^2 = R^2$$ for $|x|, |y| \leq R$ and for $|z| \leq R_z <R$. This creates a sphere with the top and bottom cap cut off. $R_z$ will be ~ $2R/...
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+200

A trigonometric problem when calculating distance to the boundary of a convex hull

Suppose we have a sphere and a point outside of the sphere. We denote the point outside as $v$ and the origin of the sphere as $x$. The convex hull of the sphere and $v$ should be like an ice cream ...
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Finding solution to Calculus of Variation of linear functional whose domain consists of vector valued function

Problem Statement: Find $x^*$ such that it solves the optimization problem $$\max_{x \in \Omega} \quad f(x) = e_i^TAx$$ $$ \Omega = \{x: t \to \Delta^{n}|x \in C^1, x(0) = x_o\}$$ Where $\Delta^...
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69 views

Find maximum value of a function [closed]

$a$, $b$, and $c$ are real numbers, and $a+b+c=0$ and $a^2+b^2+c^2=2$. I need help finding the maximum value of: $$\big|a^2b^2(a-b)+b^2c^2(b-c)+c^2a^2(c-a)\big|$$ To be honest, I don't know where ...
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For which point of $x+y+3z+k=10$, the expression $x^2+y^2+9z^2+4k^2$ is minimal? [closed]

For which $x,y,z,k \in \mathbb R$ of $x+y+3z+k=10$ is the expression $x^2+y^2+9z^2+4k^2$ minimal?
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how to minimize $tr (X^T W - Y^T W_p)(X^T W -Y^TW_p)^T$ in closed form

Assume we are dealing with matrices. Then how to minimize $$ E(W,W_p) = tr (X^T W - Y^T W_p)(X^T W -Y^TW_p)^T $$ w.r.t both $W, W_p$ simultaneously? I can calculate the derivatives of $W$ and $W_p$ ...
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Scheduling grid optimization

I am trying to optimize the programming of multiple TV channels for a given week. For each show (a day, a time and a TV show) it is possible to forecast in advance the number of people that will watch ...
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2answers
41 views

Maximizing a convex quadratic function in CVX and Matlab

I understand that a convex function can not be maximized as there is no such value. However, consider the following function: $$\begin{array}{ll} \text{maximize} & 3x^2 + 5y^2\\ \text{subject to} ...
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related to biconcave optimization

I have a bivariate function $f(x,y)$ both $x,y$ can assume values within closed interval i.e. $x_1\leq x\leq x_2$ and similarly $y_1 \leq y \leq y_2$. I know that for a fix value of $x$ the function ...
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2answers
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L1 minimization problem with nested sums as LP problem

I've been trying to solve this problem but I have an issue with the fact that there is a sum under each absolute value. I'm trying to convert this minimization problem (with respect to $x, y_1, \dots,...
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1answer
36 views

Why isn't Linear Programming less convoluted? [Soft Question]

Just a quick question. So I'm taking a course in linear optimization, and one of the things that we're going over obviously is the simplex method. I just started the class so I may not be seeing the ...
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29 views

Linear program with ceiling or floor functions

Is it possible to solve a linear program where constraints have ceiling or floor functions applied to variables (with maybe some constants)? For instance: $$\lceil (x_1 + a)/b \rceil + \lceil (x_2 + c)...
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Convex optimization with $\ell_0$ “norm”

I have an optimization problem of the form $$\begin{align*}\text{minimize }\;&f(x)\\ \text{subject to }\;&||x||_0 \le t,\end{align*}$$ where $t$ is a given constant and $f:\mathbb{R}^d \to \...
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Prove this property of the Hessian

I have been reading about the hessian for a scholar work about optimization and I find this property: Let be $H_{P_0}$ the determinant of the hessian matrix for the Lagrangian function $\mathscr{L}(x,...
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Working with Lagrange multipliers, reducing gradients is okay, right?

I am employing the method of Lagrange multipliers to determine a maximum. As part of this, I arrive at the following equation involving two gradients and the parameter $\lambda$, as is common for ...
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Finding dual of certain problem

Could anyone help me finding lagrangian function and lagrangian dual of the following problem: \begin{equation} \begin{split} \max_{X}\quad & \operatorname{trace}(H X H^T)\\ \text{s.t} \quad &...
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Economics maximization problem linear activity

Consider the vectors: $a_1 = \begin{pmatrix} 0 \\ -1 \\ 1 \\0 \end{pmatrix}, a_2 = \begin{pmatrix} 0 \\ 0 \\ -1 \\1 \end{pmatrix}, a_3 = \begin{pmatrix} 2 \\ 0 \\ 0 \\ 1\end{pmatrix}$ Find a single ...
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1answer
20 views

Convex set equals convex functions within optimization?

Can optimizing a convex function subject to convex constraints be written as optimizing the function subject to a convex set? Does the intersection of convex nonlinear ineualities necessarily describe ...
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34 views

What is the easiest way to optimize the weighted sum of L2 norms?

I have the following cost function (solving for $M$ - the $x_i$s are known): minimize $\sum_i\sum_j(w_{ij} \cdot (x_i-x_j)^T\cdot M\cdot(x_i-x_j))$ ($w_{ij} \in [-1,1] $) subject to: $M \succeq 0$ (...
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When is this argmax-based function continuous?

Let $w: \mathbb{R}^+\to \mathbb{R}$ be a continuous, strictly-increasing and strictly-concave function. define the following function: $F: \mathbb{R}^+\times\mathbb{R}^+ \to \mathbb{R}$: $$F(s,t) = \...
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Proof: $\underset{\|q\|=1}{\max} q^TAq = \lambda_{\max}$ with $q$ the corresponding eigenvector ($A$ symmetric)

This problem is quite old and there should be similar problems. I know the following technique: \begin{equation} \begin{aligned} q^TAq=q^TU\Lambda U^Tq=(U^Tq)^T\Lambda (U^Tq) \end{aligned} \...
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1answer
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Advantage of multi-objective optimization over single objective

What are the advantages of multi-objective optimization over single objective? I am specifically thinking about MO and SO in Genetic Algorithm. I have surfed the net and found many articles talking ...
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1answer
41 views

Linear programming with a product term in the objective function

The title might sound a little weird. I actually want to ask if this problem can be solved as a LP. And if so, how to convert the product term? set $P=\{1,2,3,\ldots,n\}$ for index $i$. Variables $...
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108 views
+150

Two Approaches Two Different Solutions: Optimal Controls vs. Different Method

If I try to solve a problem two different ways, I get two different answers which generally means I am committing some horrible sin! Given the problem, \begin{align} \min_u\ S &= \int dt\ L(x, u) ...