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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Finding extrema with multiple constraints without Lagrange multipliers

Find the maximums and minimums of $z = 15x+14y$ with constraints $0 \leq x \leq 10, 0 \leq y \leq 5, 3x+2y \geq 6$ I obviously can't take the partial derivatives of inequalities, so I'm at a loss ...
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Lagrange Multiplier inconsistent

max $f(x,y,z)$ subject to $x+y+z=1$ $$f(x,y,z)= - (x+0.5y)\log(x+0.5y) -2(0.25y+0.5z)\log(0.25y+0.5z)-\log(2)(1.5y+z)$$ $$\frac{\partial F}{\partial x} = - (1+\log(x+0.5y))$$ $$\frac{\partial F}{\...
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How to find center and radius of hand-drawn circle? [duplicate]

You are given a set of points {(X1,Y1), (X2,Y2),...} which represent a hand-drawn circle, so it's not perfect. You are asked to find the center and radius of this circle. My intuition tells me this ...
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can I get help in solving this equation using simplex method big-M method

Objective: $\max Z= 100x_1+300x_2+400x_3$ s.t. $10x_1+20x_2+30x_3≤1600$ $\;\,\quad10x_1+15x_2+20x_3≤1500$ $\;\,\quad x_2+x_3≤50$ $\;\,\quad x_1+x_2+x_3=70$ $\;\,\quad x_1,x_2,x_3≥0$
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Maximize $\sum_i \mathrm{rate}_i$ s.t. $\mathrm{rate}_i$

Question related to optimization problems. $$\mathrm{maximize} \sum\limits_{i=1}^{M}\log\left(1+f_i(\mathbf{x})\right)$$ $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;\;\mathrm{...
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43 views

If $A + B = \frac{\pi}{3} (A,B>0),$ Then the minimum value of sec A + sec B is?

If $A + B = \frac{\pi}{3} (A,B>0),$ Then the minimum value of sec A + sec B is? I know the condition for minima but here there are two simultaneous variables , how and with respect to what do I ...
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422 views

The minimum distance from the circle $x^2+(y+6)^2=1$ to parabola $y^2=8x$?

What are the coordinates of the points on the parabola $y^2=8x$ which are at the minimum distance from the circle $x^2 + (y+6)^2=1$?
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85 views

Optimize function on $x^2 + y^2 + z^2 \leq 1$

Optimize $f(x,y,z) = xyz + xy$ on $\mathbb{D} = \{ (x,y,z) \in \mathbb{R^3} : x,y,z \geq 0 \wedge x^2 + y^2 + z^2 \leq 1 \}$. The equation $\nabla f(x,y,z) = (0,0,0)$ yields $x = 0, y = 0, z \geq 0 $ ...
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115 views

KKT point of a constrained optimization problem

Min$_{x}~x$ Subject to $x \geq 0$ For this problem, is $(x^{*}, \lambda^{*})=$$(0,0)$ a KKT point ? My try : I formulated corresponding Lagrangian and tried to find out the KKT point(s). ...
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93 views

Lagrange multiplier - Find maximum on surface

I need someone to walk me through a 3 variable lagrange problem, since I haven't been able to find a reliable source to teach me, please. Here it is: Find the maximum of the function $F(x,y,z) = 2x+...
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69 views

Microprocessor pinout problem. How many pins do I need for given N buttons?

At the begginning I'd like to sorry for the long intro, but I think it helps in fully understanding the problem. Of course the math problem is much shorter, so if you are not interested in the full ...
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44 views

Classification of critical point

The critical point of this function are $(0,0),(-1/3,-2/9),(-1/3,2/9)$. And for $(0,0)$, I get the difference that $△f=f(0+a,0+b)-f(0,0)=a^4 + (3a+1)b^2$ will always greater than zero where a,b are ...
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What are active constraints?

I am asked why the constraint $x_1\leq 2$ would be active when maximizing $$ 8(x-1)^2 +2(y-1)^2 $$ subject to $$12x+12y=126$$ But I am not sure what it means for a constraint to be active. We are ...
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Understanding the shape of $\phi''(x)=F(\phi(x))$

Hello I've got a question and no idea to get a solution. Maybe someone can give me an advice. The following problem is given: There is given a function $\phi \in C^2([a,b])$. Furthermore there is a $...
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48 views

How do you call the following iterative solving method

I have the following implicit equation $$ x= f(x) $$ which I solve by starting with some value for $x$, then setting $x$ to the new value $f(x)$ and so forth until convergence. How is that method ...
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559 views

What minimizes the Chebyshev Distance?

For an arbitrary number of dimensions, I know that the mean minimizes the distance using the L2 norm and that the geometric median minimizes the distance function using the L1 norm (though I have yet ...
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74 views

T-shaped polygons

Is there any coefficient that can indicate T-shaped polygons ? Examples of T-shaped polygons:
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368 views

How do I setup the lagrangian for this problem?

I have a function $y(x)$, that I would like to maximize, subject to two constraints. It is given by: $$ \max_{x} \ y(x) = a \ cos(x) + b \ sin(x) \\ \text{subject to:} \\ x \geq 0 \\ x \leq \frac{2}{...
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636 views

Find the maximum or minimum value of the quadratic function by completing the square.

Find the maximum or minimum function of the quadratic function by completing the squares. State the value of $x$ at which the function is maximum or minimum. $y=3x^2+7x+9$ I already posted similar ...
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216 views

points on surface that is closest to the point

I need to Find the maximum and minimum values of $$f(x, y, z) = x + 5y + 2z$$ on the sphere $$x^{2} + y^{2} + z^{2} = 1$$
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698 views

Why is this weighted least squares cost function a function of weights?

Here is a picture from my book regarding weighted least squares: Totally lost here, so I extracted the main nested issues confusing me: First Question: I know that in any LSE we want to minimize ...
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What is the Minimal cost?

A power house, $P$, is on one bank of a straight river $W$ meters (m) wide, and a factory, $F$, is on the opposite bank $L$ meters downstream from $P$. The cable has to be taken across the river, ...
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392 views

Positive, Negative definite and indefinite matrix

A symmetric matrix is positive definite iff all eigenvalues are positive. I have been given a 3X3 symmetric matrix. I have calculated the eigenvalues two of which are negative. Does this mean this ...
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48 views

Convex Function Help and Counterxample

Given $g: \mathbb{R}^n \to \mathbb{R}$ is convex and $f:\mathbb{R} \to \mathbb{R}$ is convex and increasing. Show that $(f \circ g): \mathbb{R}^n \to \mathbb{R}$ is convex. I had no problem proving ...
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78 views

Is it possible to reduce a lambda expression to it's smallest equivalent form?

In the Untyped Lambda Calculus, is it possible to reduce any arbitrary expression to it's smallest equivalent form? (defined as an expression with the smallest number of lambda terms) If so, is there ...
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Maximum area under a curve by calculus of variations

I am asked to find the function that has the maximal area for a given length L when x runs from -a to a. I calculated the integral to be varied as follows: $$ \int_{-a}^{a}\ y + \lambda \sqrt{1 + (\...
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49 views

Extrema on a given set

Could you tell me if my approach to finding extrema on a set is good? Let's take a function $f(x,y,z)=x+y+z$ and a set $N= \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 \le z \le 1 \} = \{(x, y, z) \in \...
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190 views

Finding minimum of the trace of the matrix equals finding maximum of the trace of the inverse matrix?

Let $K$ be a positive definite, symmetric matrix. Let $C$ be a nondegenerate matrix of the same order. Elements of $K$ and $C$ depend on some parameter $a.$ Is it true to say that $$ \min_{a}{trace\...
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609 views

Sum of weighted squared distances is minimized by the weighted average?

Let $x_1, \ldots, x_n \in \mathbb{R}^d$ denote $n$ points in $d$-dimensional Euclidean space, and $w_1, \ldots, w_n \in \mathbb{R}_{\geq 0}$ any non-negative weights. In some paper I came across the ...
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887 views

Maximum area of a rectangle inside a triangle

I recently came across a problem where it gave a triangle with integer side lengths, and it asked you to find the maximum area of a rectangle of a triangle. I solved the problem correctly, but it ...
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488 views

Constrained maximization of Leontif utility function $\min(x_1, x_2)$

The maximization problem is: Maximize $u(x_1, x_2) = \min[a_1x_1, a_2x_2]\; \ \text{s.t.}\;\; p_1x_1 + p_2x_2 \leq$ $w$, in which $x_i, p_i$ is the amount and price of good $i$, $w$ is the ...
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305 views

Optimization calculus Problem

I have a calculus final coming up and was going over a practice exam my professor gave me and I came across a problem I was struggling with. I would post a picture but I am having trouble posting a ...
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Optimization Problem

Can someone please help me with this minimization problem? I dunno what to do after replacing p(x) with given s.t.
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Perimeter equation

A window is in the form of a semicircle surmounted over a rectangle. Thew total perimeter of the window is 12m. Note:This is a part of a maxima minima question that I was trying to solve. I could ...
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189 views

What is the difference between projected gradient descent and ordinary gradient descent?

I just read about projected gradient descent but I did not see the intuition to use Projected one instead of normal gradient descent. Would you tell me the reason and preferable situations of ...
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Relaxed optimization problems

The original problem is \begin{align} \min & f(x) \tag{1}\\ \text{s.t.} & \text{constraint 1} \tag{2}\\ & \text{constraint 2} \tag{3}\\ \end{align} However, it is very hard to deal with ...
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191 views

nth root algorithm: value of initial guess?

I wonder what value one would choose to maximize efficiency to make an initial guess for the nth root algorithm (supplementary constraint: only with the five operations: +, -, *, /, % (integer modulo))...
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Why does determining the nature of local extrema for $\mathbb R \to\mathbb R$ functions require twice continuous-differentiability?

In the text Elementary Classical Analysis, why does Marsden specify the condition "twice continuously differentiable" here? Isn't mere twice-differentiability sufficient for the purpose indicated? I'...
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Derivatives question help

The question is :Find the derivative of $f(x)=e^c + c^x$. Assume that c is a constant. Wouldn't $f'(x)= ce^{c-1} + xc^{x-1}$. It keeps saying this answer is incorrect, What am i doing wrong?
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Minimize the area of a triangle

Let $A \neq B$ be fixed points outside a fixed circle with centre $C$. The point $D$ can be chosen freely on the circle. The goal is to minimise the area of triangle $ABD$. Degenerate triangles (...
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Optimisation of a rectangles area under a function curve

I have a questions asking for the dimensions of the rectangle with the largest area that has two bottom corners on the x axis and two top corners on the curve $y=12-x^2$. I have plotted the curve and ...
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''min $c^tx$ subject to $x^tAx=1$'': is is possible to solve it with Lagrange multiplier or in the scope of KKT?

I find a problem: Minimize $c^tx$ subject to $x^tAx=1$, where $A$ is a positive semidefinite symmetric matrix. But the question obligates to use KKT but I am trying to apply simple Lagrange ...
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Optimization question

A rectangular beam will be cut from a cylindrical log of diameter 1m. For part a) I have shown that the beam of maximal cross-sectional area is a square. Then 4 rectangular planks will be cut from ...
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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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197 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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578 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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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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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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Why compare f(n)/f(n-1) = 1 to solve for the maxima of a discrete function?

I am aware of a general strategy where you have a discrete function, e.g., $f = \dfrac{{10 \choose 5}{n-10 \choose 15}}{n \choose 20}$ And in order to find the maximum, you solve $\frac{f(n)}{f(n-1)}...
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87 views

Optimal Configuration for a Set of Points

Consider a set of $n$ points on the plane with positions $\mathbf{p}_1,\dots,\mathbf{p}_n$, such that each point $i$ has at least one neighbor $j$ at a distance of no more than $\lambda$ away from it (...