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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Absolute Conditional Constraint in MIP

I have seen approaches for both Absolute Constraints (linear programming : Absolute value in constraint in mathematical model) and Conditional Constraints (Integer Programming Conditional Constraints)...
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Generate function from data

I have a series of inputs and outputs : Inputs -> Outputs 1,2,3 -> 4 4,5,6 -> 5 7,8,9 -> 6 Is there a field of study that can generate a single ...
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Is the CMF of a log-concave PMF also log-concave?

If a PDF is log-concave, then its CDF is also log-concave. The proof I know for this uses the derivative of the log function, see Proposition 1 in this paper. Does this also hold for discrete ...
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Is there a better way to find the closest point on a line?

I'm given a question that asks: "Find the point on $L(x) = 4x-3$ that is closest to the point $(1,3)$." My best guess was to find the derivative of the distances and set it equal to zero and solve to ...
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MILP how to make constraints numbers as a decision variable

I am trying to build a MILP. I need to set the number of linear constraints in the model as a decision variable. For example: ...
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Calculate lesser value that can take the side c=? [on hold]

EDIT: Consider a right triangle , it is satisfied that: $ab + bc + ac = 100$ Determine the smallest value that the side $c$ can take (without brute force)
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Could anyone explains the solution for constrained optimization in a paper “Embedding a semantic network in a word space”?

I'm reading a paper "embedding a semantic network in a word space". In the paper, a problem for embedding word sense is formalized as bellow. $$ \text{minimize}_{E,p}\sum_{ijk}w_{ijk}∆(E(s_{ij}),E(n_{...
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Optimal Apple Eating Strategy

You hate apples. As a result, you have angered the apple king and are being punished. You will have to eat $n$ apples before the apple king is willing to let you leave. The apples are marked from $1$ ...
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4answers
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Why must the determinant of the hessian of a scalar function be positive for there to be a local min/max? Intuition needed

Is there any intuition behind having the determinant of the Hessian matrix being negative corresponding to a saddle point, and positive corresponding to a max/min depending on the sign of $f_{xx}$ for ...
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18 views

Best fit curve through several line segements

I have a question about creating a best fit curve that incorporates several lines on a graph. To give some background, I am working on the geometrical design of a solar reflector. The reflector is ...
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16 views

Optimize an Trace matrix form

In paper " Generalized Low Rank Approximations of Matrixces the Dimension of matrix are follow: $A_i$ is $r$ x $c$ L is $r$ x $l_1$ R is $c$ x $l2$ $D_i$ is $l_1$ x $l_2$ why it says ...
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Lagrangian fuction for optimization problem

I have an optimization problem \begin{equation}\label{eq:optimi_joint1} \begin{aligned} \text{minimize}_{\mathbf{w_p}\in \mathbb{R}^M,\mathbf{\Lambda_p}\in \mathbb{R}^{M\times M}} \ \ &\kappa \...
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2answers
111 views

What is the optimal route for visiting Pokéstops in Pokémon Go?

Okay, I've got a fun problem for you, which was not suited for the gaming stackexchange: Pokéstops are GPS locations with a certain radius. When you are in the radius, you can get certain ingame ...
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0answers
36 views

What would the derivative of this objective function be?

My question comes from image processing community, In our Machine Learning algorithm, we have a predicted value $D$ and its equivalent ground truth $D^*$ where their difference is: $d_i=D-D^*$. (...
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1answer
24 views

Trace minimization problem with “block diagonal diagonal” constraints?

I've reduced my optimization problem to the following trace minimization problem: $$\min_X\text{tr}(AXB),$$ subject to that $X$ is a block diagonal matrix whose blocks are all the same -- a diagonal ...
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2answers
30 views

On the maximal of polynomial at a point

I faced this problem when I studied polynomial. Let $p(x)=ax^3+bx^2+cx+d$ be a cubic polynomial with real coefficients, and $p(5)+p(25)=1906$. Find the maximal value of $|p(15)|$. I ...
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37 views

Linear programming optimization problem

I need some hint, where I can find programming algorithm for next optimization problem. I need to write a code to solve some system of equations with several restrictions. Let's assume, we have $N$ ...
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22 views

Maximum point of a modulus function

For our project on Maxima and Minima of functions, we have to do functions of type $\frac{k}{|x-a|+|x-b|}$. So, I chose $f(x)=\frac{2}{|x-1|+|x-2|}$ I noticed that the derivative is positive for $x&...
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Optimal way to partition sum of linear equitations

Suppose we know the values of $x_i$, $i=1,2,3,...,n$. How do we choose an integer $k$ such that $\sum_{i=1}^{k-1} (k-1-i)x_i + \sum_{i=k}^{n} (n-i)x_i$ is minimized; $0 \leq k \leq n$? My initial ...
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37 views

maximising sinusoidal functions

I have come across a maximisation problem that I do not know how to handle. I have posted the question here in the past. I have the following function to maximise for $x,y$ $$f(x,y)=a_1 \cos(x) +b_1 ...
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Solving simple LP problem with Lagrange multipliers

Hi just as a test I'm trying to solve the following LP with Lagrange multipliers. $min -x_1$ $s.t$ $x_2 \leq 1 - x_1$ $x_1, x_2 \geq 0 $ I add slack variables to have a equality constrained LP ...
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2answers
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Evenly filling spaces for a specific average value

Imagine I have $N$ spaces. Each space can be empty, or occupied. Given a fixed point value $x$ between zero and one, I would like to evenly populate the $N$ spaces such that $\frac{N_{\text{occupied}...
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Optimisation: mayer, lagrange and bolza problem

Regarding the determination of an optimal curve, can someone please help me and define the mentioned problems in words, i.e. when to use what. I know, that a bolza problem is the combination of mayer ...
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33 views

Confusion of a formula about Lagrangian

Recently, I am reading a paper about eigenvalue problems. Consider the following problem, which occurs at the first page of the paper. \begin{align} \text{minimize}\quad &x^TAx \\ \text{subject ...
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Equivalence between standard optimization problem and Langragian form

Given a problem: $$\min_x f(x)$$ subject to $$g(x) \le C$$ In general, when it is equivalent to the problem $$\min_x f(x) + \lambda g(x)$$ for certain $\lambda$? Here my equivalence means : the ...
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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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1answer
51 views

Meaning of $Ax \leq b$

I continue to come across $Ax \leq b$ or $Ax= b$ in optimization problem, but I am having trouble interpreting the meaning of this. Does this have a similar meaning to the following (Cramer's Rule) ...
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20 views

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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39 views

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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1answer
78 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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32 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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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 $f'(x)\ge1$ [closed]

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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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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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
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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
22 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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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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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
38 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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14 views

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
71 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 ...