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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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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12 views

Armijo conditions vs Reduction Conditions in Non-Linear Line Search

Overview Line search typically consists of four stages: Direction: Search direction Initial Step Size: length to search along the line on the first sub-iteration Bracket: find an interval along the ...
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algorithm to find the root of a real-valued function $f$

I see in a book the following algorithm to find the root of a real-valued function $f$ $$ \theta_{n+1} = \theta_{n} + \epsilon f(\theta_n); \epsilon >0 $$ with the condition that the initial ...
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15 views

Quadratic programming using Python

guys I'm trying to solve quadratic programming problem with constraints. I know how to solve simple quadratic problems using scipy.optimize like following: Define objective function as F = ...
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147 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
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In a given set, does there always exist a parameterization $(f_1,f_2)$ of a curve such that $\frac{f_2f_2'}{f_1f_1'}=-1$?

Suppose we consider a level set $S$ (set for which a function is constant) of a function from $\Bbb R^2$ to $\Bbb R$. Suppose there is a curve lying in $S$ and parameterized by a continuously ...
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47 views

minimizing sum of different least squares?

Can we write the minimization problem: $$\operatorname{min}\limits_{x\in\mathbb{R}^n}\sum_{i=1}^{n}\|C_i x-b_i\|_2^2$$ as a least square problem?
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3 views

Equilibrium in a first-price common value auction with a normal distribution?

Does anyone know of a paper that derives equilibrium bidding strategies in a first-price common value auction when signals are assumed to come from a normal distribution?
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1answer
61 views

Finding Shortest distance between a Sphere and Ellipsoid?

Suppose that ,I have a Sphere and an ellipsoid as Sphere: $(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2 = R_1^2$ Ellipsoid: $\large\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$ ...
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25 views

How to deal with non-existent derivatives in Lagrangian?

I am stucked at a detail in a constrained optimization problem: Question Assume that the objective function is continuous on its domain $D$, but at some points $Z \subseteq D$ it is not ...
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20 views

Maximising a function under a constraint

Let $$f(x,y,z) = 4x+2y+5z^2 \text{ and } A=\{(x,y,z) \in \mathbb{R^3} ; \, x^2+y^2+z^4 \leq 5 \}.$$ Find the maximum of $f$ on $A$. My question is the following: How do I prove that the maximum must ...
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23 views

Coordinate descent with constraints

Coordinate descent is a powerful method for solving optimization problems like $$\min_x \tfrac{1}{2}x^T A x + b^T x + \lambda ||x||_1$$ where $A$ is symmetric and positive definite, $\lambda>0$ ...
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34 views

Optimization Software

I am currently trying to optimize an equation that contains 4 variables. It is nonlinear and non-convex. In mathematica, to optimize this equation, I take the partial derivative of each variables, ...
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37 views

How to find a line that minimizes the average squared perpendicular distance from the given points to the line?

I have set of points scattered around the origin. How to find a vector, such that the average squared distance (perpendicular distance) from points to the vector is minimised? Added For example, ...
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561 views

The minimum and the maximum of $y=\sin^2x/(1+\cos^2x)$

I was asked to find the minimum and maximum values ​​of the functions: $y=\sin^2x/(1+\cos^2x)$; $y=\sin^2x-\cos^4x$. What I did so far: $y' = 2\sin(2x)/(1+\cos^2x)^2$ How do I check if ...
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76 views

Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
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240 views

Business Linear Programming Question

Now I don't need you guys to do my homework for me; however, I am a little stumped Xara Stores in Canada imports the designer-inspired clothes it sells from suppliers in China and Brazil. Xara ...
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Biggest subset of $\{1, 2 … 1000\}$ such that difference between any pair of elements $\neq 4, 7$

The problem, as stated in the title, is to find the maximal size of a subset $V$ of $S = \{1, 2, ... 1000 \}$ such that no two elements of $V$ have a difference of 4 or 7 between them, i.e. $x \in V ...
3
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2answers
146 views

How many routes possible in the traveling salesman problem with $n$ cities? And more…

SO the general answer I come across on the internet is $(n-1)!/2$. But it would seem to be $n!$, or at least $(n-1)!$. Which one is it? If you have 2 cities, you would have 1 path. So $(n-1)!/2$ ...
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Ask a question about an example in a course note on optimization problem with equality constraint

I have two difficulties on understanding the solution to an example in a course I took this semester on optimization. This example is given to illustrate the usage of Lagrange multiplier method ...
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Objective function: $\min \frac{1}{2}\sum_{t=t_0}^{\infty} y_t ^2$

Objective function: $\min \frac{1}{2}\sum_{t=t_0}^{\infty} y_t ^2$ $y_t=E_t y_{t+1}-0.5(i_t-2)+u_t$ where $t=t_0,t_1,\dots$ $u_t=0.5 t_{t-1}+\epsilon_t$ where $\epsilon$ is i.i.d(independent and ...
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Baseline predictors model

I implemented baseline predictors model (like it is told in Recommender systems handbook pp 148-149). b_ui = mu + b_i + b_u where mu is overall average rating ...
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699 views

How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
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65 views

Sum of two polyhedra is a polyhedron

I'm reviewing for a midterm next week in an optimization course. Currently, I'm having a great deal of trouble with a review problem. The problem is as follows: Let $P$ and $Q$ be polyhedra in ...
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20 views

Alternative to Hungarian Algorithm to determine minimum cost?

Is there a graphic calculator (CAS technology) method to solve minimum cost problems/allocations that are normally completed with the Hungarian Algorithm... Hungarian Algorithm is time consuming, ...
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27 views

resources about sparse global constrainted optimization

Please recommend a good resources (books/articles/software) about sparse global constrained optimization?
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Critical points, minima and maxima of a constrained optimisation problem

$f(x_1,x_2,x_3) = x_1 x_2 + x_2 x_3 + x_3 x_1$ with the constraint $x_1+ x_2 + x_3 = 1$. Now I test the critical points by taking $\nabla f = 0$, hence at $(x_2 + x_3, x_1 + x_3, x_2 + x_1) =0$. This ...
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How to load warehouse pallets efficiently?

Assume that we would wan't to develop a warehouse management system, which picks up plastick boxes and stacks them on a pallet. A pallet has a maximum of 5 vertical box stacks and the maximum height ...
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Find the point where the function $C(t) = 0.24t/(t^2 + 5t +4)$ attends its maximum [closed]

The concentration, $C$, in micrograms per deciliter, of a drug in a patient's bloodstream is given by the equation, $C(t) = \dfrac{0.24t}{t^2 + 5t +4}$ , where $t$ is the number of hours after the ...
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Are there some functions that cannot be optimized using calculus?

I've been working on a project to maximize a functions output using a genetic algorithm. However, from the limited calculus I know I thought there were methods to find the maximum of a mathematical ...
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squaring the equality constraints

When creating an unconstrained optimization problem from an equality constrained one, the usual way to build the Lagrangian, is by adding a term consisting of a multiplier, multiplied by the equality ...
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Closest Positive-Definite Matrix Subject to a Contraint

Given a positive, semidefinite, real 2n by 2n matrix $A$, is there a formula or an algorithm that finds the closest (in some sense, preferably Frobenius distance) positive, semidefinite, real 2n by 2n ...
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What is the minimum of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ over all complex numbers?

Find the Least value of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ My try:: Let $A(2,2)$ and $B(-1,5)$ and $C(6,-2)$ and $P(x,y)$ be a point Here $A,B$ and $C$ are the point ...
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Deriving estimators for the parameters a and b that minimize the random error - setting up linear regression variables?

I'm reviewing old notes, and I know I solved this way back when, but can't remember how to know: Consider the simple linear regression model: $$Y_i = a + bX_i + \epsilon_i$$ where $Y_i$ is the ...
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Minimizing the cost of production by choosing between two options with different resource demands

Suppose that the firm has two possible activities to produce output. Activity $A$ uses $a_1$ units of good $1$ and $a_2$ units of good $2$ to produce $1$ unit of output. Activity $B$ uses $b_1$ units ...
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Steepest descent direction with surface constraint (geometry problem)

Let say I have a function $f(x,y,z)$, defined on a surface by the level curve $g(x,y,z)=c$. I want to know what is the direction of the steepest descent at a given point, taking into account the ...
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Total number of subsets

I am trying to solve this hackerrank problem https://www.hackerrank.com/contests/101jul14/challenges/colorful-polygon. Not able to understand the editorial. Can some explain how to solve it using ...
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Optimization problems: Finding the optimal path

I'm still trying to get the hang of optimization problems in calculus and I'm looking for a little help. I'm having trouble finding equations to model the following problem: I'm fairly sure I need to ...
2
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2answers
64 views

Why does this vector derivation hold?

I have the following variables/matrices: $$A \in \mathbb{R}^{m \times n} , \quad p \in \mathbb{R}^{n}, \quad \Sigma \in \mathbb{R}^{m \times m}, \quad w \in \mathbb{R}^{m}$$ where $\Sigma$ is a ...
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Find the maximum or minimum value of the quadratic function.

Find the maximum or minimum value of the quadratic function by completing the squares. Also, state the value of $x$ at which the function is maximum or minimum. $y=2x^2-4x+7$ $x^2$ has a coefficient ...
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Solver for least squares

I'm looking for a numerical solution to the constrained least squares problem below: $$ \min_\mathbf{x}\|\mathbf{a+Bx}\|^2 ~~\text{s.t}~~\|\mathbf{x}\|^2 \leq \alpha^2$$ where $\mathbf{a} \in ...
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Finding gradient of an objective as a PDE

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
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Relation between girth and lower bound of the number of vertices

Let $G$ be a graph with girth $g > 3$. Suppose that every vertex in $G$ has degree at least $k > 1$. Can we found a nice lower bound for $|V(G)|$? Let me be more specific on what I want: ...
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If the Jacobian matrix is positive definite, does that imply that the optimization problem has a unique solution?

My PhD adviser told me that if the Jacobian matrix of the optimality conditions is positive definite, then it implies that the optimization problem has a unique solution. I was wondering what is the ...
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2answers
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Find an equation for a moving rod

The two endpoints of a 1-metre long rod have an initial position at $(0,0),(0,1).$ The rod slides continuously to the position $(1,0),(0,0)$ sweeping out a region in the positive quadrant. Determine ...
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1answer
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How can I find the point (X, Y, Z) which minimizes this quantity?

I have a number of equally powerful light sources $L_i, 1 \le i \le N$ at points within a cube $(x_i, y_i, z_i), -1 \le x_i, y_i, z_i \le 1$. The intensity of each light falls off with distance ...
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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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Is this a discrete time Lyapunov function?

I have an algorithm to optimize a process. It is a discrete time algorithm. Every iteration of this algorithm changes the state of the process. I found a function, say $f(s)$, where $s$ is the state ...
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Trace minimization subject to constraints

I have seen in an article that $ \min_{\mathbf{K}} \hspace{0.2cm} tr[\mathbf{K} \Sigma \mathbf{K}^T]$ s.t. $ \mathbf{KH} = \mathbf{I} $ where $\mathbf{H}$ is of full column rank yields, ...
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60 views

Least squares with a quadratic inequality constraint

Is there a closed form solution for the following least squares problem: $$ \min_\mathbf{x}\|\mathbf{a+Bx}\|^2 ~~\text{s.t}~~\|\mathbf{x}\|^2 \leq \alpha^2$$ where $\mathbf{a} \in \mathbb{C^{M\times ...