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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How to get the closed form solution of a non-convex optimization problem?

I want to know if there is a closed form expression for the optimal objective function? How can I get it, if it does exist? Condition: $h,f\in \mathbb{C}^{N\times1}, \epsilon > 0 $. $\max \ \ ...
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411 views

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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95 views

Limit of a function at its maximum

Let $f_n(x,y)$ be a real function in the domain $0\leq x\leq 1$ $0\leq y\leq 1$ I would like to compute $A = \lim_{x \to 0}\left( \arg \max \limits_{y} f(x,y) \right)$, The problem is that it ...
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554 views

(1)Maximize the trace of a matrix. (2)Minimize the trace of its inverse matrix. Are (1) and (2) equal?

I need to solve an optimization problem which aims to $\text{maximize}$ $\text{Trace}[(U^HU)^{-1}]$. The diagonal elements of $U^HU$ are all positive. I want to know whether it is equal to solve the ...
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464 views

Minimum number of lines covering n points

Let there be n points in the plane. I want to know the minimum number of horizontal and vertical lines covering all the points in the plane. My initial approach started like this, 1) for each point I ...
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2answers
154 views

How to prove this max absolute value equation?

How to prove this equation? $$\max(|x_1-x_2|,|y_1-y_2|) = \frac{\left|x_1+y_1-x_2-y_2\right|+\left|x_1-y_1-(x_2-y_2)\right|}{2}$$
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77 views

Optimization of wave drag formula ( gradient calculation of double summation)

i want to ask a question about gradient calculation of double summation term wave drag formula The formula (objective function to minimize !) shown above calculates the wave drag of an aircraft, S is ...
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Maximize $W(x) - (\ln(x) - \ln{\ln{x}})$

How can you maximize $f(x) = W(x) - (\ln(x) - \ln{\ln{x}})$ for $x\geq 2$? Numerically the answer seems to be at around $x \approx 41$ where you get $f(41) \approx 0.31$. Mathematica suggests the ...
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16 views

How to test non-negativity of linear functions on homogeneous semi-algebraic sets?

Fix a homogeneous semi-algebraic set $S \subset \mathbb{R}^{n}$, by which I mean that the set $S$ is defined by inequalities $f_{1},...,f_{k} \geq 0$, where $f_{1},...,f_{k}$ are homogeneous ...
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How to solve linear system of equations in 1 and inf-norm?

I have the problem to find a linear program that is equivalent to solving the problem that finds a minimum for $||Ax-b||_1$ and $||Ax-b||_{\infty}$. We defined a linear program as follows: $min_{x} ...
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38 views

finding range of function of three variables

Three real numbers $x$, $y$, $z$ satisfy the following conditions. $x^{2}+y^{2}+z^{2}=1~$, $~y+z=1$ Find the range of $~x^{3}+y^{3}+z^{3}~$ without calculus. I solved this problem only with ...
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138 views

Maximize two-variable linear function

How would you maximize the following function (with integer domain) $$f(x,y) = a * x + b * y$$ subject to $$c * x + d * y \leq N$$ $$x \geq 0, y \geq 0$$ the constants $a, b, c, d, N$ are known ...
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0answers
34 views

Simplifying a difficult expression to input it in R

The function is $\Large f(X,Y|\mu_1,\mu_1,\theta)=\frac{\phi (X-\mu_1)\phi (Y-\mu_2)\theta(1-e^{-\theta})e^{-\theta (\Phi(X-\mu_1)+\Phi(Y-\mu_2))}}{[1-e^{-\theta}-(1-e^{-\theta \Phi ...
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2answers
76 views

Minimisation of a distance sum

I have a list $L$ of $N$ numbers, and I want to choose $k$ numbers $\{x_1,x_2, \ldots,x_k\} \subseteq L$ in such a way value $S$ of the those K numbers is minimum. $$ S = \sum_{0< i < j <= k} ...
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How $(P_{r+1}-P_r)$ is maximized?

This is from DeGroot's "PROBABILITY and STATISTICS"(Second edition)(Cf. pages 87 to 92).I am rewriting the relevant stuff. Let $r$ be a positive integer and $r\leq n$. ...
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71 views

Minimum moves to transform a list to another?

Given two list of n positive elements. We are allowed to perform only one transformation which is to increment each element of the list except one. What are the minimum number of transformation ...
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377 views

How to determine wether critical points (of the lagrangian function) are minima or maxima? [duplicate]

$f(x,y) = 2x+y$ subject to $g(x,y)=x^2+y^2-1=0$. The Lagrangian function is given by $$ \mathcal{L}(x,y,\lambda)=2x+y+\lambda(x^2+y^2-1), $$ with corresponding $$ \nabla \mathcal{L}(x,y,\lambda)= ...
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401 views

Find the minimum,maximum, infimum and supremum of sets?

If $X$ is the intersection of all the intervals $(1-\frac{1}{n^2},1+\frac{5}{n^3}]$ for $n=1$ to infinity, what is the minimum, maximum, supremum and infimum of $X$? If $Y$ is the intersection of all ...
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For each of the following functions, what do the first- and the second-order optimality conditions say about wether 0 is a minimum on $\mathbb{R}$.

For each of the following functions, what do the first- and the second-order optimality conditions say about wether 0 is a minimum on $\mathbb{R}$. $f_1(x)=x^2$ $f_2(x)=x^3$ $f_3(x)=x^4$ ...
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42 views

how to minimize this convex function?

$x_i$ and $y_j$ are variables. I intend to minimize this function and obtain the optimal value of $x$ and $y$: $\begin{align} ...
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159 views

Applying the Lagrangian function to find critical points

So I have the following function $$ f(x,y) = x^2+y^2 $$ subject to $$ g(x,y) = x+y-1 = 0. $$ And I have to use the Lagrangian to find the critical points, and determine wether they are ...
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138 views

What's a really good book for a course titled “Optimization and Control Theory”?

I can't seem to find one that shows a lot of examples with the theory. Could I get some help? Also, it would be a bonus if the book/material is readily available online so I can download it onto my ...
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313 views

Determine all the extrema of a function subject to a non-linear constraint.

QUESTION Determine all extrema of the function $$f(x,y) = x+ 2y $$ subject to $$x^2 + y^2 - 80 = 0$$ ATTEMPT I don't think I understand what I'm supposed to do. This was in a test and I ended up ...
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709 views

How many n square can fit into a square of side N

Suppose we have n small squares of equal sizes that has area w. Suppose we have a fix square S of area A such that for area A, one area w < area A. If square S's area A, length, and width are ...
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273 views

Minimize $\|Ax-b\|$ where $x$ is a binary vector

For a software project I'm involved on, I have a situation where I have a large vector that is the sum of some smaller vectors. I know all the possible small vectors, and I know that no two of them ...
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189 views

This question is a basic optimization problem, also a linear algebra question:

Let $p$ be a direction of unboundedness for the constraints $$Ax = b, x ≥ 0.$$ Prove that $−p$ cannot be a direction of unboundedness for these constraints.
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Geometric interpretation of a critical point, i.e. of $q(t) := f(x + t(y-x))$.

So, I know what critical points are. But hear me out on the following notes I made: For $x,y\in \mathbb{R}^n$ we define $$q(t) := f(x + t(y-x)), $$ then $$q'(t)=\nabla f(x+t(y-x))^T(y-x).$$ Now, if ...
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Can someone help me please?

$A^+A=I$ where $A^{+}=(A^TA)^{-1}A^T$, $A_{m \times n}$ I have tried with $(A^TA)^{-1}=A^{-1}{A^T}^{-1}$ but the matrix is not squared
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34 views

For any positive integer $n$, what is the value of $t^*$ that maximises the following expression?

For any positive integer n, what is the value of t* that maximises the following expression? $$\displaystyle \sum_{j=1}^{n-t^*}\left(\frac{t^*-j+2}{t^*+j}\right)$$ where $t^*$ is some integer in ...
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29 views

On a quadratic optimization

The problem is formulated as follows: Given $0\neq x \in \mathbb{R}^n$, and $k\leq n$, consider the following optimization problem $$\min_{\textrm{rank}(C)=k}x^t(I_n-C)^t(I_n-C)x$$ where $I_n$ be the ...
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Sum-to-one constraint

This is a general question, but I am asking it since I am not able to find any good material online. Can someone please explain what's meant by a "sum-to-one constraint"? Thanks.
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455 views

Distance between point and sine wave

I have a project where I need to know the exact minimal distance between a point $(e, f)$ and a sine wave $y = a + b\cdot\sin(cx+d)$ Is there any way of calculating this? If not, is there a way to ...
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54 views

Add and subtract in optimization

I have a problem related to the use of the "add and subtract" strategy in optimization problems. This is related to a question I asked on Cross Validated. I got really helpful answer to this question, ...
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345 views

Equivalence of Maximizing Products, Sums, and Sum of Logs

Throughout this question assume $f_i \ge 0 \forall i $. I know that for any (single) function the following is true $$f(x^\star) \ge f(x) \text{ }\forall x\in X$$ iff $$\log(f(x^\star)) \ge ...
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Second-order Lagrange condition

I am not sure what the second-order Lagrange condition is and how it applies to this? Minimize $x^2 + y^2$ Subject to $x^2 - y - 4 \leq 0$ and $y - x - 2 \leq 0$. Please can someone assist me in ...
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How to find the maximum of $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a+b}+\dfrac{1}{a+c}$ given certain constraints.

Let $a,b,c\ge 0,$ and such $a+b+c=1$. Find the maximum of: $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a+b}+\dfrac{1}{a+c}$$ My try: ...
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43 views

Proving certificate of inequality

I have a question about proving the certificate of inequality in the given question: If there exists $y$ such that $y^T A \leq 0$ and $y^T b < 0$, then $Ax = b$, $x \leq$ 0 has no solution. I ...
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55 views

Inequality Constrained Optimization Problem

I am working on the question displayed below. I am not sure if I understand it correctly and I am looking for some input. So, I am asked Why is $x^*$ a local maximum for $f$ subject to the set ...
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139 views

Reference request - second derivative test for function of two variables that includes details of what you can infer when discriminant is zero

The second derivative test for functions of two variables as I have learned and taught in calculus classes says, in part, that if at a point $D=f_{xx}f_{yy}-(f_{xy})^2$ is zero then we can tell ...
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Attain minimum on boundary of a convex set?

It is well known that there exists a unique minimum norm vector over a closed convex set. Suppose we have a Banach space X (if it needs to be more concrete we can think of $L_2$, the space of square ...
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Minimization of $\text{tr} (W^TMW)-\text{tr}(NW)$ subject to $W^TW=I$

Is there a closed-form solution for finding W that minimizes the objective function: $\text{tr} (W^TMW)-\text{tr}(NW)$ subject to $W^TW=I$ where $M$ and $N$ are fixed matrices. I find it difficult to ...
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1answer
164 views

Projectile Trajectory with Air Drag

Given the following equations and values, Find an initial theta value to maximize horizontal range with air drag. $f(x)=\tan(\theta)*x-16+(x/(200*\cos(\theta))^2$: height with no air drag ...
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Multi objective optimization into single objective.

I read that it is possible to convert a multi-objective optimization problem into single objective by using weighted sum method. I wanted to know if it is a good idea to convert a two objective ...
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1k views

Zero Eigenvalues for Hessian Matrix

I need to show that along any line passing through the origin, $$F(x,y) = 3x^4 -4x^2y + y^2$$ has a minimum at $(0,0)$ but that without the restriction, there is no local minimum at $(0,0)$. The ...
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214 views

Maximizing a ratio of convex matrix functions by minimizing a difference?

Given that $g(.),h(.)$ are twice-differentiable convex quadratic real functions whose domain is the set of all real matrices while the range is the set of positive real numbers, then: Is maximizing ...
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For the function $f(x) = x^2 \sin(1/x)$ for $x \ne 0\,$and $f(x)= 0$ if $x=0$, show that $f$ does not have local maximum or minimum in $0$.

We have $f(x) = x^2 \sin(\frac{1}{x})$ for $x$ does not equal $0$ and $f(x)= 0$ if x equals $0$. I know that the function is continuous for all $x$ by using the squeeze theorem. I also know that in ...
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Optimization with inequality constraints

Could someone give a solution to this problem as well as an interpretation of the results? I am not sure how to deal with inequality constraints. Part 1: Find the minimum value of $f(x) = |x|^2$ ...
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how to solve this equation containing “min”?

$$ xy=128,x+y=\min $$ How to find $x$ and $y$ with the minimum sum? This example is simple and can be done by brute forcing but I want to know what is the proper way of solving it. How to solve ...
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58 views

Find minimal $\alpha_3$ such that $u\in H^3(\Omega)$ and $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$

My instructor presented me the quiz below but forgot to define key terms such as minimality and $H^3$. Quiz Let $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$. Find the minimal $\alpha_3$ such that $u\in ...
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31 views

Minimum of function

Let $f:\mathbb{R}\to\mathbb{R}$ be a function with $f(x)=\dfrac{(x^4-2ax^3+3a^2x^2-2a^3x+a^4+9)}{(x^2-ax+a^2)}.$ Determine the minimum of the function, if we know, that $-2\leq a\leq2$, $a\neq0$. I ...