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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Transpose of matrix inverse: $(AA^T)^{-1}A^Tb \stackrel{?}{=} (A^TA)^{-1}A^Tb$

Given the matrix equation: $$ x^TA^TA = b^TA $$ I'm trying to find the least squares solution (i.e.; trying to minimize $r=||Ax-b||$). The matrix $A$ is not necessarily symmetric. When I solve it ...
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340 views

Derivation of Euler-Lagrange equation

Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation. If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is $\dfrac{\partial ...
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Revised Simplex: reduced cost and related constraint

In the revised simplex method, you can get the reduced costs straightforward from the tableau. I know which they are, but I don't know which reduced cost I should "relate" to which constraint. Will ...
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2answers
119 views

find maximum and minimum for any function

I'm writing an optimization algorithm thats supposed to find the maximum and minimum value of any given function. Whats the fastest numerical approuch to do so?
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706 views

How to calculate the hessian of a matrix function

I am estimating a model minimizing the following objective function, $ M(\theta) = (Z'G(\theta))'W(Z'G(\theta))$ $Z$ is an $N \times L$ matrix of data, and $W$ is an $L\times L$ weight matrix, ...
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What is going on with this constrained optimization?

I'd like to figure out what is going on when trying to maximize a function (below $a_i$ are real numbers) $F = a_1a_2 + a_2a_3 + \cdots + a_{n - 1}a_n + a_na_1;$ When we have active constraints ...
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4answers
240 views

Maximum surface inside a triangle

If I have a triangle with sides of length a, b, c and I have a rope of length L, what is the maximum surface of a boundary I can form with that rope that is entirely inside the triangle. Normally, ...
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2answers
496 views

genetic algorithm binary encoding

I am trying to write a program for maximizing a function using a genetic algorithm. The function has $n$ integer variables $x_1 \dots x_n$, such that each variable is in the range [-n,n]. What is ...
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105 views

Maximizing a convex function

The following problem is exercise I.6 from Bellman's Dynamic Programming. Consider the problem of maximizing the function $$ F(x_{1} , \ldots , x_{N}) = \sum_{i = 1}^{n} \varphi(x_{i}), $$ subject to ...
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How to project points in 3-space into a 2D subspace while minimizing the maximum change in Euclidean distance?

We have a small set of points in $\mathbb{R}^3$ (around 4 to 10 points, say). I would like to project these points onto a 2D subspace such as to minimize the maximum change in Euclidean distances. ...
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Understanding the Hamiltonian function

Based on this function: $$\text{max} \int_0^2(-2tx-u^2) \, dt$$ We know that $$(1) \;-1 \leq u \leq 1, \; \; \; (2) \; \dot{x}=2u, \; \; \; (3) \; x(0)=1, \; \; \; \text{x(2) is free}$$ I can ...
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Minimum ladder over wall optimization

A fence 6 feet tall runs parallel to a tall building at a distance of 2 feet from the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to ...
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1answer
182 views

Epigraph of a function f: D $\rightarrow$ R is convex iff epif(f) is a subset of D*R which is a convex set

As in the topic, how to show that $epi(f)$ is convex iff $epi(f)$ belongs to D*R which is a convex set.
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1answer
87 views

Strict local minimiser

Let $\Omega$ be a convex subset of $R^n$ adm f is a real valued, twice differentiable function. Let $x^*$ to be a point in $\Omega$ and suppose that there exists $c \in R \, c >0$ s.t. for all ...
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201 views

What is the dual of this optimization problem?

Consider the points $x_1, \ldots, x_N \in \mathbb{R}^n$, and a (locally bounded, convex) function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. I am looking for the dual of the following optimization ...
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1answer
2k views

Optimization: Minimize cost of pipeline

A small resort is situated on an island off a part of the coast of Mexico that has a perfectly straight north-south shoreline. The point P on the shoreline that is closest to the island is exactly 6 ...
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Optimization and distance (minimum time)

A small island is 5 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 2 miles per hour and can walk 3 miles per hour, where should the ...
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942 views

Minimum distance between $x = -y^2$ and $(0,-3)$

Find the minimum distance from the parabola $x + y^2 = 0$ (i.e. $x = -y^2$) to the point $(0,-3)$. This is a homework question. When I try to use the derivative and substitute $-y^2$ for $x$, I ...
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894 views

Optimization and window area

A Norman window has the shape of a rectangle with a semi circle on top; diameter of the semicircle exactly matches the width of the rectangle. Find the dimensions of the Norman window whose perimeter ...
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224 views

Optimization and Rent

The manager of a large apartment complex knows from experience that 110 units will be occupied if the rent is 342 dollars per month. A market survey suggests that, on the average, one additional unit ...
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1answer
1k views

Optimization and fence size

A fence is to be built to enclose a rectangular area of 250 square feet. The fence along three sides is to be made of material that costs 6 dollars per foot, and the material for the fourth side costs ...
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191 views

Facets of the convex hull as solution of an optimization problem?

Given $N$ points $x_1, x_2, ..., x_N \in \mathbb{R}^n$, consider their convex hull $$\mathcal{C} = \text{conv}( \{ x_1, ..., x_n \} ) = \bigcap_{j=1}^{J} \{ x \in \mathbb{R}^n : \ A_j x \leq b_j \} ...
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1answer
145 views

What is the minimal isoperimetric ratio of a polyhedron with $5$ vertices?

I'm asking and answering this question to provide a partial answer to this question and a comment on this answer at MO. The isoperimetric ratio $\mu$ of a solid is the ratio $A^3/V^2$, where $A$ is ...
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Constrained optimization max $ f(x,y) = x+y$ subject to $x^2+y^2 \leq 4, x \geq0, y \geq0$

max $ f(x,y) = x+y$ subject to $x^2+y^2 \leq 4, x \geq0, y \geq0$ I need to solve this by the Kuhn Tucker conditions without using concavity of the Lagrangian.
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Optimization. I need help finding the maximum profit

Josh wants to start a cell phone repair business. Josh determines that $x$ phones can be repaired daily at $p$ dollars per repair, where $x=175-p$. The cost of repairing $x$ phones per day is ...
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1answer
623 views

hessian matrix not positive definite at a minimum?

I have a function for which I want to find the global minimum. The function is: ...
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1answer
69 views

Optimizing the area of a triangle in space.

A triangle has two corners, $(8,0,3)$ and $(0,8,3)$ and a third curve in space that consists of all points $(8,8,a^{2}+3)$, where $a$ is a real number. Calculate the area of the triangle as a function ...
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102 views

Formulation of a problem as semidefinite programming

I would appreciate some help with this problem: $R$ is a positive semidefinite matrix $\in{R}^{n\times n}$, $A \in{R}^{n\times m}$. I need to formulate this optimization problem as semidefinite ...
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1answer
45 views

$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta$ ,$\theta=$?

This question is a follow up question to this answer. In the equation: $$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta.$$ $a$ and $b$ are given. What is the best way to solve for ...
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Find $\operatorname{argmax}_x \operatorname{corr}(Ax, Bx)$ for vector $x$, matrices $A$ and $B$

This is similar to, but not the same as, canonical correlation: For $(n \times m)$ matrices $A$ and $B$, and unit vector $(m \times 1)$ $x$, is there a closed-form solution to maximize the correlation ...
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3-D Absolute Max/Min over closed&bounded region

Find the absolute max and min values of $f(x,y)=2x+y^2-2$ on the closed and bounded region that lies outside the upper half-circle of $\{(x,y)| x^2+y^2=1\}$, and inside the rectangle given by ...
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SDP relaxation of non-convex QCQP and duality gap

Short version Is there a duality gap between a QCQP problem and the SDP problem obtained through lagrangian relaxation? A paper I'm studying is using this fact, but I cannot achieve the authors' ...
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791 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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Find maximum of the function $f(x)=\dfrac{e^{\frac{2x}{x+1}}-1}{x}$

Let $x\ge0$. Find maximum $$f(x)=\dfrac{e^{\frac{2x}{x+1}}-1}{x}$$ I think this maximum is $2$, I hope this problem have some nice solution,Thank you
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Finding the max/min distance from an ellipse to a line (Lagrange Multiplier Method)

An ellipse is specified $ x^2 + 4y^2 = 4$, and a line is specified $x + y = 4$. I need to find the max/min distances from the ellipse to the line. My idea is to find two points $(x_1, y_1)$ and ...
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Use of low rank approximation of a matrix

I am trying to figure out why do we need a low rank approximation of a matrix. Why is it used and where? Any insights?
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Textbooks on modern optimization (on machine learning) with exercises

I know Boyd's famous Convex Optimization, but for me it's a little bit old because it was written in 2003 and some progresses have been made during this decade. The book Optimization for Machine ...
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Is it possible to calculate the following optimization expression?

I'm faced with this expression: $$max(xy) : x \leq a, \; y \leq b,\; c = \frac{x}{y}$$ where all variables are greater than or equal to zero. I know the values for a, b, and c but I need to find ...
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1answer
95 views

Infimum of a multivariable function

How to find the infimum of following equation over y. $$f(x,y)=x^T Ax+x^T By+y^T Bx+y^T Cy$$ $$inf_y f(x,y)=?$$ where A and C are symetric matrices
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On norm selection for the solution of an overdetermined linear system

I am considering the following linear system: $Ax = b$ Where: $A$ is $9000 \times 139$ $x$ is $139 \times 1$ and sparse $b$ is $9000 \times 1$ Most of the resources I have found online point to ...
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1answer
1k views

Find maximum of integral

I am trying to find the maximum value of $$I=\int_0^y \sqrt{x^4+(y-y^2)^2}\,dx$$ for $0 \leq y \leq 1$. At $y=1$ the value $I=1/3$ which I think is the answer. How can you prove this?
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Maximize $f(x) = x^3-3x$ subject to constraints

I would like to understand more about how to maximise functions of one variable subject to constraints. How can you find the maximum value of $f(x) = x^3 - 3x$ subject to $x^4+36 \leq 13x^2$? The ...
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Transformation of the maximum of a product

I am studying for an exam and I can't quite understand how to obtain the result below. $$\max_{x\in [x_0-h,x_0+h]}|(x-x_0 + h)(x-x_0)(x-x_0-h)| = h^3\max_{u\in [0,1]}|(u^2-1)u|$$ I know how carry ...
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Example of a function on $[0,2]$ with no maximum of minimum

Can anybody help me by providing an example of a function (or the graph of such a function) defined on $[0, 2]$ but with no maximum and no minimum? An explanation is also appreciated. Context Such a ...
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Dynamic Programming Trouble, Optimizing time

A robot goes from terminal to terminal collecting bolts. The robot needs to collect at least $m$ bolts and there are $n$ terminals. Terminal $i$ gives the robot a certain number of bolts denoted by ...
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156 views

minimization problem on differential equations - optimal control

I am trying to minimize an time-integral of a linear function with respect to differential equations. The problem is formally defined as follows: Given $\lambda< \mu_1, \mu_2$ fixed ...
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3answers
709 views

How to find maximum and minimum volumes

I would appreciate if somebody could help me with the following problem: Q: Let $S$ be the region bounded by the curves $y=\sin x \ (0 \leq x \leq \pi)$ and $y=0$. Let $V(c)$ be the volume of the ...
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70 views

Optimization of Integral

$F(x,y)=\int_a^{b} \int_c^{d} G(x,y,s,t)dsdt$. I wish to find $x$ and $y$ (subject to some constraints) that minimize $F$. $F$ is not expressible in closed form. Is the only way to solve this problem ...
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How to find the minimum of $f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$?

I need to find the minimum of $f(x)$ with $$f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$$ Could you help me with some clues?
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1answer
101 views

Quadratic Forms and Newton's Method

Consider the function $f(x,y) = 5x^2 + 5y^2 -xy -11x +11y +11$. Consider applying Newton's Method for minimizing $f$. How many iterations are needed to reach the global minimum point? Why should ...