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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What is the name of this problem? linear Matrix equation optimization?!

I have almost no knowledge in linear algebra but I need to understand the process of solving a problem. In fact I'm looking for some keywords or hints to know what exactly should I be Googling! So any ...
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46 views

Logistic regression for football results - Estimating coefficient through maximum likelihood

Consider two football teams $V$ and $L$ with strengths $W_V$ and $W_L$, respectively. Let's assume that the draw probability $\mathbb{P}(Draw)$ is known. Then this model is supposed to give estimates ...
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14 views

Multi-objective optimization or single objective optimization?

I have this function: A(x)= P(x) / B(x) Firstly I thought about doing an multi-objective optimization, maximizing A(x) and minimizing B(x) because this two values are very important. But if I just ...
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4answers
82 views

Minimize the function $f(y_1,y_2)=3 y_1^2+8y_2^2$ [closed]

I would like to minimize $f(y_1,y_2)=3 y_1^2+8y_2^2$ with the constraints $g(y_1,y_2)=y_1^2+y_2^2=1$. I thought I could use the Lagrange multipliers, but it is not work. Is there anyone could show me ...
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98 views

Maximizing a linear function over an ellipsoid

Let $A \in \mathbb{R}^{n\times n}$ be a positive definite matrix, $x \in \mathbb{R}^n$ and $c \in \mathbb{R} \setminus \{0\}$. I got to determine the maximum $$\max\{c^Ty:y\in \mathcal{E} (A,x)\}$$ ...
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34 views

Change of variables in minimization

I have the following non linear programming to solve: $$\left\{\begin{matrix} \min & (x-y)^2 +e^z+e^{-z} \\ \text{s.t.} & xz=0 \\ & yz=0 \end{matrix}\right.$$ The book suggests to ...
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1answer
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Does projected gradint descent(pgd) results in the same minimizer as the one given by unconstrained gd and projected back on the constrained set?

For $f: \mathbb{R}^n \mapsto \mathbb{R}$ with $f(x) < \infty,\;\forall x \in \mathbb{R}^n$ and for convenience let's assume $f$ is continuously differentiable. Suppose we are trying to solve the ...
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54 views

How can I experiment with Lagrange multiplier in QCQP?

Suppose we want to solve following optimization problem (it is a PCA problem in this post) $$ \underset{\mathbf w}{\text{maximize}}~~ \mathbf w^\top \mathbf{Cw} \\ \text{s.t.}~~~~~~ \mathbf w^\top \...
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1answer
50 views

A question about Lagrange multiplier in optimization

I read @amoeba 's answer in this post, PCA optimization problem is $$ \underset{\mathbf w}{\text{maximize}}~~ \mathbf w^\top \mathbf{Cw} \\ \text{s.t.}~~~~~~ \|\mathbf w\|_2=1 $$ where $\mathbf C$ ...
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0answers
12 views

KKT conditions for different inputs.

So I have the following problem: I'm trying to get a demand function for a nonlinear 2 variable optimisation problem. There are 3 inequality constraints. Doing the usual thing I get the following ...
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45 views

Why the original MINLP and Linearized MILP are giving mismatched results?

I have an MINLP and its linearized formulation problem given below where the objective (nonconvex) and constraint C4 are nonlinear. We linearized them by applying some known techniques. However, when ...
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24 views

Optimal number of operations in the given scenario?

Suppose $$A_1=\{x_1+x_2+x_3,\quad x_2+x_3+x_4,\quad x_3+x_4+x_5\} \\ A_2=\{x_0+x_1+x_2, \quad x_0+x_1+x_8, \quad x_0+x_7+x_8\} \\ A_3=\{x_{10}+x_{11}+x_{12}, \quad x_{11}+x_{12}+x_7, \quad x_7+x_8+x_{...
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34 views

Extremas on multivariable functions

if the gradient Of a function f(x,y,z) has all its partial derivatives (fx, fy) at a point p equal to zero but the partial derivative z at that point is equal to a constant i.e fz= 12. In this case ...
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3answers
59 views

Distance from a set to a point

There is this exercise I cannot understand well. It asks me for the distance between this set in $\mathbb{R}^3$ $$U = \{(x, y, z)\ |\ ax + y - 2z = 0, z = 0 \}$$ and the point $(0, b, 1)$. Also it ...
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24 views

Book Recommendation for Infinite Dimensional Stochastic Optimization Problem in Discrete Time

Let $X(k)$ be i.i.d. discrete random variables and for all $k=0,1,2,...,N-1$, let $X:=(X(0),X(1)...,X(k))$ and $f := (f(0), f(1),...,f(k))$ with $f(k)$ be the decision function at time $k$, I want to ...
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2answers
34 views

Minimising a Loss Function requiring Matrix by Matrix Derivative

I am trying to minimise the following cost function with respect to $X$: $\mathbf{C}(X) = ||{M \cdot X \cdot \mathbf{1}_{N \times 1} - T}||_{2}^{2}$ Here, $M$, $X$, $T$ are matrices of dimensions $a \...
2
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1answer
90 views

Given a population of fish with exponential growth, what is the optimal strategy for fishing?

Suppose we have a population of fish, say $10000$, with an exponential growth each year of $30\%$. If we want to collect as many fish as possible in, say 10 years, a natural question to ask is: ...
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36 views

Minimizing trace with equality constraints

I would like to solve the following trace-minimization under equality constraints optimization problem: $$W^* =\arg\min \operatorname{Tr}[WCW^T] \text{ s.t. } A=B^TW^TWB$$ where $W,C\in\mathbb{R}^{...
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1answer
48 views

Minimizing a strictly convex function with inequality constraint

So we've been learning about the Kuhn Tucker conditions in my non-linear optimization course and I've been having trouble with this problem: QUestion: description here Question: a strictly convex ...
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35 views

Failure of second derivative test of two variable function where the all the partial derivatives are equal

Actually I am looking to find the local minimum of the following function : $$F(x,y)=\frac{\Gamma(x+y+1)\Gamma(n-x-y+1)}{\Gamma(n+1)}$$ The partial derivatives of this function are: $\begin{align} ...
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Individually checking constraints for convexity in Optimisation problem valid?

I have a quadratic minimisation problem where both the objective fn and constraints have some quadratic terms. (Such as a throttle variable (continous) * On/Off (integer variable)). My question is: ...
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Upper bound for $\gcd(a,b)$ if $\frac{a+1}{b}+\frac{b+1}{a}\in\Bbb{N}$

Suppose that $a,b$ are two positive integers so that $\frac{a+1}{b}+\frac{b+1}{a}$ is also a positive integer.Find the best upper bound for $\gcd(a,b)$. My work: $\frac{a+1}{b}+\frac{b+1}{a}=\frac{...
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1answer
104 views

Maximizing area of a pentagon

Suppose $a,b,c,d,e$ are pairwise distinct positive integers. Consider a pentagon with sides $a,b,c,d,e$ and with angles maximizing its area (we assume that a pentagon with such sides exists). It is ...
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1answer
29 views

Minimization of vector sum through rotation

I'm curious if there's an algorithmic way to find the minimal vector sum of $N$ vector magnitudes by applying a rotation $\Phi$ to each individual vector. As an example in two dimensions, if I have ...
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1answer
27 views

Robusness of median

If we let $X$ be a set of pints in $\mathrm{R}^2$, and let $g(X) = \arg \min_{y \in \mathrm{R}^2} \sum_{x_i \in X} \parallel x_i -y \parallel_2$ (geometric median of $X$). If $X$ and $X'$ are ...
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3answers
62 views

Maximum und Minimum [closed]

I need help for c) also I solved a and b, but c) I could not. Let $f~:~\Bbb R^2\to \Bbb R$ be defined as $f(x,y)=e^{-x^2-y^2+2x-2y}$ a) Determine all local extrema for $f$. b) Determine all ...
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Optimization of function on integer hypertetrahedron

I have the following optimization problem: Let $k,n \in \mathbb{N}$ with $k < n$. Let $N:=\{1, \dots,n\}$ and $D := \{^{t}(x_1, \dots, x_k) \in N^k \vert \sum_{i=1}^k x_i = N\}$ (hypertetrahedron)...
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1answer
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IF statement as Linear Constraint

I am writing a linear program, but I am currently having troubles writing a certain constraint, which is basically an IF-statment. I will try to explain it as detailed as possible: IF: $x_{it'}(t' +...
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Extrema on (compact) vinculum

My textbook ask to find the extrema of $f(x,y) = 2x^2+y^2$ on $x^4-x^2+y^2-5=0$. It uses the lagrangian multipliers to find critic points.. Then it computes the function on these points then says "...
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Optimal decomposition of discrete function into sum of factorised terms

I am trying to solve the following optimisation problem. Let $x_i \in \{1, \ldots, N_i\}$ be discrete variables, and $f(x_1, \ldots , x_n)$ any real-valued function. I want to decompose $f$ into a ...
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9 views

Is it possible to convert a loss function minimization problem into an eigenvector problem?

I have a very vague feeling that the problem of optimizing a loss function is related to the problem of finding the smallest/biggest eigenvector. However, I don't have the expertise to see this ...
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Optimized placing of same-size squares into rectangles

Suppose that we have several squares of the same size. We want to draw n rectangles (red and yellow rectangles here) to contain these squares. The goal is to have ...
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Solving Equations involving max operation

I would like to know how to solve this set of equations for v*(h) and also v*(l) Assume all other variables are known..concentrate on the 3rd equality in case of v*(h)..the first 2 are not needed. I ...
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36 views

Minimizing a function including max functions

Consider the following problem. Let $\mathcal{N} = \{1,2,\ldots,N\}$ and $\mathcal{N}^i = \mathcal{N}\setminus \{i\} $. For each $i \in \mathcal{N}$ and for each $S \subset \mathcal{N}^i$, we have a ...
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Second-order stochastic optimisation - proof of convergence.

Consider the following scheme. We want to minimise some function $f(\theta) : \mathbb{R}^n \rightarrow \mathbb{R}$. We have access to noisy estimates of the first order derivative $g_t$ such that $E[...
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modelling a composite objective function (max + argmax) as an (integer) linear program

Suppose $\mathbf{x} = [x_1, x_2, \ldots, x_n]$, where $x_i \in \{0, 1\}$ are binary variables. We know for a fixed $\mathbf{w}$ the following problem is an Integer Linear Program: $$ \arg\max_{\...
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3answers
108 views

Minimizing $\cot^2 A +\cot^2 B + \cot^2 C$ for $A+B+C=\pi$

If $A + B + C = \pi$, then find the minimum value of $\cot^2 A +\cot^2 B + \cot^2 C$. I don't know how to solve it. And can you please mention the used formulas first. What I can see is that if one ...
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1answer
16 views

Is the EM-algorithm the same thing that variational inference in LDA?

I am new in the probabilistic topic modeling, and I need to understand deeply the LDA process, I understand what want to do the inference process in LDA, and I understand too that there is 2 "types" ...
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Properties of polyhedron solving constrained max problem

This is a question for people who don't have trouble to think in more than two dimensions. Don't hesitate to ask clarifying questions! Let us suppose we have $n$ random variables $X_i$ that are iid ...
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What is meant by a function being linear in two variables?

I'm trying to understand the Mangasarian condition in the context of dynamic optimization (see here p 8.12) and am not sure what exactly is meant by a function $f(x,u)$ being linear in $x$ and $u$. If ...
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linearize average success probability constraint

In my optimization problem, I've a constraint to calculate the average success probability of a path. $x_{i,j}$ is binary variable defined as: $$ \begin{align} \label{eq3:1} x_{i,j} = \begin{cases} ...
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35 views

Construct set of basis vectors with most zeros

I have a set of basis vectors that describes a linear subspace. The following is an example of three 16-dimensional vectors: $$\quad\quad\!\! \left( \begin{array}{c} b_1 \\ b_2 \\ b_3 \end{array} \...
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Tuning schemes for additional valves on a brass instrument

So this question is very much about music, but it's entirely about a mathematical part of music, and that part is tuning and how it relates to pipe length in a brass instrument. Background: I am ...
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18 views

What is the difference between these two Bellman equation?

Bellman equation: $V(x) = max \{F(x,y)+ \beta V(y)\}$ $V(x) = max \{F(x,y), \beta V(y)\}$ When to use the plus and when to use the comma? Do they get the same form of sequence problem? Would you ...
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convexity of 3 variables minimax problem

Here is the function $f(x)=\underset{y\,\in \,C}{min}\quad \underset{z\,\in \,\mathcal{K(x,y)}}{max} \quad z^TQz$ in which $\mathcal{K(x,y)}:=\{z\,|\,x=y+z^a-z^b\}$ $x,y,z \in C$ $C \subseteq R^...
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20 views

Potentially A Double Optimization Problem

I am trying to figure out how to handle this optimization problem, but I am having difficulty in processing the calculation. Consider $E$ to be the set $E = \{(x,y) \in \mathbb{R}^2 | x^2 + y^6 = 2\}.$...
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1answer
38 views

System of equations with multiple answers, but only natural numbers (and other constraints)

Okay so basically I want the solution with the smallest sum when you add up the integer variable solutions that give a solution to: $975a + 880b + 790c + 585d + 487e + 440f + 292g + 260h + 530i + ...
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1answer
36 views

Convex or non-convex function

I want to minimize the following function $$\frac{a}{bxy+cd}e^{\frac{a}{bxy+cd}}H+2-\Gamma(1,\frac{eaf}{b(1-x)},\frac{eagf}{bx(1-y)})$$ where $a,b,c,d,e,f,g,H$ are constants and greater than $0$. $\...
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35 views

Maximum of Contour Line

Consider the potential $U(x,y)=ay^{2}+b(e^{x-y}-1)^{2}+c(e^{x+y}-1)^{2}$ where $a$, $b$ and $c$ are known constants. I want to move through a contour line of this potential $U(x,y)=k$, say $y=g(x)$. ...
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1answer
24 views

Algorithm for scheduling event observers

I'm reviewing different algorithms to solve a scheduling problem and was hoping someone with a better breath in the area might help me focus on the right class of algorithms. Basically the problem is ...