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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Mixed Interprogramm remodeling

for example i have the following problem min z 5 x_1a + 6 x_1b - 3 x_2a + 0 x_2b <= z -3 x_1a + 0 x_1b - 1 x_2a + 2 x_2b <= z x_1a + x_1b = 1 (Constraint say of this group only one variable ...
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The meaning of $\lambda$ in Lagrange Multipliers

This is related to two previous questions which I asked about the history of Lagrange Multipliers and intuition behind the gradient giving the direction of steepest ascent. I am wondering if the ...
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104 views

Gradient and Swiftest Ascent

I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$) The standard proof is to note that the ...
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33 views

Can someone explain Method of Lagrangian multipliers

Can someone explain Method of Lagrangian multipliers to a beginner? I need some knowledge about solving problems using this method. If someone can provide the basic details in a simple manner along ...
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68 views

Infinity norm minimization

I am wondering how to minimize an objective function of the following form: $$\min_{\mathbf{x}\in\mathcal{R}^{MN}} \|\mathbf{x}-\mathbf{y}\|_\infty + \lambda\mathrm{TV}(\mathbf{x})$$ Here, ...
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Maximizing Expectation

A bag contains $b$ balls in total, $r$ of which are red, while the rest are white. In a game a player removes balls one at a time from the bag (without replacement). He may remove as many balls as he ...
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47 views

First derivative test and uniqueness of local extrema

This is the context in which my question lies. See below for the actual question. Let $f(x)$ be differentiable everywhere and have a minimum at $x^*$. Then for every $x$ in a proper neighbourhood ...
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64 views

Determine the smallest number P

I have here a hard problem, which I couldn't solve. Denote $M$ the set of all functions $f:[0,1]\to\mathbb{R}$ with the following properties: $f(x)\ge0, \forall x$ in $[0,1]$, $f(1)=1$, $f(x+y)\ge ...
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Euler Lagrange equations

I need to minimise $$\int\limits_\Omega|\nabla H_\epsilon(\phi)|\,dx\,dy$$ with respect to $\phi$. Where $H_\epsilon$ is the regularised Heaviside function, so that it is differentiable. This can be ...
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46 views

proving a theorem of alternative

I've read the following exercise in my book: Let $A\in\mathbb R^{m\times n},b\in\mathbb R^m,c\in\mathbb R^n$. Then exactly one holds: $Ax=0,c^t\cdot x=1$ with $x\geq0$ has a solution $A^ty\geq c$ ...
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545 views

Distance from Ellipsoid to Plane - Lagrange Multiplier

Find the distance from the ellipsoid $x^2 + y^2 + 4z^2 = 4$ to the plane $x + y + z = 6$. I'm trying to do it using Lagrange multipliers over the distance equation, but then it just gets ...
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Optimization with intervals

I am trying to solve a specific problem, and I was able to summarize it in the following optimization problem. I have a portfolio comprised of two assets. Asset 1 has return $r_1$, standard deviation ...
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53 views

How to combine Unitary Matrices in a clever way?

I am trying to implement genetic-type algorithms on unitary matrices. Hopefully I should be able to use this question for the mutation part. But I am having an issue with the cross-over step. So here ...
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54 views

Minimization of product function subject to constraints

I want to minimize the following function: $\prod_{i=1}^{n}{x_i}$ Subject to the following constraints: $\sum_{i=1}^{n}{x_i}=1.1+(n-1)(0.1)$ and $0.1 \leq x_i \leq 1.1$ How should I go about it? ...
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Steepest Descent for a Quadratic

I have the formula for calculating the step-size for steepest descent for a given quadratic. However, the formula says that Q is positive definite. My Q is not. Does the same formula apply? Admins, ...
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37 views

Non-convexity of an energy functional

How would I go about showing that the following Mumford Shah functional is not convex? $$E_{MS}(u,C)= \int_{\Omega} |u_{0}(x,y) -u(x,y)|^{2}\ dx\ dy + \mu \int_{\Omega \backslash C}|\nabla ...
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217 views

Absolute extrema of a multivariable function bounded by an ellipse

I have a function $f(x,y) = 2x + x^2 + y^2$ bounded by the ellipse $x^2 + 4y^2 \leq 24$ I know how to determine the extrema within the ellipse by getting the partial derivatives and setting them to ...
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126 views

Platonic solids and charged particles

It is known that there are five Platonic solids: If, lets say, there are 4 particles with the same electricity charge and whose movement is constrained to be on a sphere, resulting forces will ...
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Minimising waste in a cutting problem.

I have three possible board sizes: $8$, $10$ and $12$ feet long. I want to make some number of cuts to these, say, $3, 2,1,1,1,6,5,3,4,2,1$ feet cuts and I want to minimize waste. I've done a quick ...
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3answers
80 views

Difficult time finding critical points using Lagrange

The function is $f(x,y,z) = xyz$ on $x^2 + y^2 + z^2 = 1$. So I have: $yz = 2x \lambda \\ xz = 2y \lambda \\ xy = 2z \lambda \\ x^2 + y^2 + z^2 = 1$ I guessed $x = \pm 1, y = 0, z = 0, \lambda = 0$ ...
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180 views

AP Calculus Help - Concave/Extrema/Concavity

A. Find all values of x in the interval (−2.6, 3.6) where f ′(x) has a horizontal tangent. B. Find all values of $x$ in the interval $(−2.6, 3.6)$ where $f (x)$ is concave upwards. Explain your ...
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49 views

Vector optimization with set constraint

This is a more generalized form of a previous unanswered question, from which I've removed all the content that wasn't relevant to the actual problem. I have a minimization problem of the form $$ ...
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124 views

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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Comparing the hardness of optimizing two similar, but different expressions

Suppose we have binary variables $y_1, ..., y_n$. To make the representation simple, we show the concatenated vector as $\mathbf{y} = (y_1, ..., y_n)$. Consider the two following functions: $$ ...
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Why is max($\frac{2}{||w||}$)= min($\frac{1}{2}$)($||w||^2$)?

I was watching a video on machine learning. The instructor says that maximizing ($\frac{2}{||w||}$)is difficult (why?) so instead we prefer to minimize $\frac{1}{2}||w||^2$. $w$ is a vector. How are ...
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Optimization Intuition

Take this question for example: What is the smallest possible sum of the squares of two numbers, $a$ and $b$, if $ab = -16$ So you get $b = \frac{-16}{a}$ and substitute. Once you find your ...
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19 views

Proving a Function is Less than a Value

I'm working on a multipart problem and I've been asked to prove that a value W < 0.5. I worked out W and reached $$W = \frac{\sqrt{X1*X2*Y1*Y2}}{X1*Y1+X2*Y2+X1*Y2} $$ but I'm not sure how to ...
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267 views

How do I convert max min problem into a linear programming problem?

Let $A$ be a given $m \times n$ matrix, $c$ a given $n$-vector, and $b$ a given $m$-vector. $$\max \min (c^T x - y^T Ax + b^Ty) \text{ such that } x,y \ge 0$$ Show that this problem can be reduced ...
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What is this problem called?

I don't know the name of this problem, but I heard it some time ago. The statement was if a planet is distance $d$ away, and you know both the current maximum velocity of spaceships $v_0$ and the ...
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52 views

Transform unconstrained optimization problems into constrained ones?

I want to formally show that the following minimization problem $$ \min_\theta||\max(0,f_1(\theta)),...,\max(0,f_n(\theta))||^2 $$ is equivalent to $$ \min_{\beta, \{w_i \}^{n}_{i=1}} ...
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Post-optimality analysis: Change in one of the constraints

Consider the LP: max $\, -3x_1-x_2$ $\,\,$s.t. $\,\,\,\,$ $2x_1+x_2 \leq 3$ $\quad \quad \ -x_1+x_2 \geq 1$ $\quad \quad \quad \quad \ > x_1,x_2 \geq 0$ Suppose I have solved the above ...
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How do I solve max min (x − y) and min max (x − y) such that y≥0 and x≥0?

solve max min (x − y) and min max (x − y) such that y≥0 and x≥0 I don't have a clue where to start.
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31 views

Maximize/minimize $1/3 x^3 + y$ with constraint $x^2 + y^2 = 1$?

I keep running around in circles when I use the Lagrangian multiplier method getting $x = 1/y$ But then when I substitute $(1/y)^2 + y^2 = 1$ I then get $1/y^2 + y^2 = 1$ and this doesn't give me ...
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80 views

Convexity of LASSO

I would like to know if some variables in design matrix are correlated then LASSO is convex or not. If you give me a proof for convexity of LASSO and ADAPTIVE lasso, I will be thankful. LASSO is ...
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Prove That the Second Moment is Minimized with a Circle Packing

Graham and Sloane studied the problem of minimzing the second moment of disks on the plane, i.e. minimize $$ U = \frac{1}{d^2} \sum_{i=1}^{n} || \mathbf{p}_i - \bar{\mathbf{p}} ||^2 $$ s.t. ...
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53 views

lagrange multiplier (with variables x,y,z)

I'm new to this topic, pls can I get hints on how to solve it: Find the point $(x,y,z)$ obeying $g(x,y,z)=2x+3y+z-12=0$ for which $f(x,y,z)=4x^2+y^2+z^2$ is minimum. Thanks in advance.
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Find absolute maximum of $f(x,y)=(x-1)^2+(y-2)^2$ within square $0\le x ,y\le 1$ (long but interesting)

Notice: *"There is another possible and common interpretation for 0≤x,y≤1, namely 0≤x∧y≤1. Hence the OP's references to the infinite. – Git Gud" If anyone ever stumbles upon this post in the ...
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490 views

Gradient-descent algorithm always converges to the closest local optima?

Assume $f(\vec x)$, which is Lipschitz continuous, has two local optima $\vec x_1^*$ and $\vec x_2^*$( $\vec x_1^*$ is the global minimum). We start the gradient-descent algorithm from $\vec x_0$ and ...
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347 views

Maximum value of $abc$ for $a, b, c > 0$ and $ab + bc + ca = 12$

$a,b,c$ are three positive real numbers such that $ab+bc+ca=12$. Then find the maximum value of $abc$
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Constrained Optimization Problem with system of differential equations

thanks in advance for any help. My question relates to a homework problem. I have been given a system of generalized differential equations with the goal of creating an algorithm that will solve ...
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56 views

Quadratic programs: is the projection onto constraints optimal?

Consider the Quadratic Program $$ x^* := \arg \min_{ x \in X } \ \{ x^\top x + c^\top x \} \ \text{ sub. to: } Ax=b $$ where $X \subset \mathbb{R}^n $ is a non-empty, convex, bounded polyhedron. ...
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Numerically optimising a sequence of matrix multiplications

I am trying to set up an optimisation problem and solving it numerically. I am still formalizing it and unsure what is the best way to solve it. It seems like a common problem, and im sure people have ...
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minimizing a function involving exponential term

Let $w\ge e$ . I want the following $$ \min_{r\geq0} r(e^r-w) $$ Is there any way to find it. Thanks.
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40 views

Halting of an algorithm

Suppose there is an algorithm that runs on a finite set. If we do not directly specify a halting condition, such as reaching a certain value, or after given number of iterations, what are the methods ...
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147 views

How can I find the vertices of a triangle by optimization?

Here is the information provided, and the hypotenuse length is minimum. How can I find the vertices of a triangle by optimization? Thanks.
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21 views

maximizing distance in a polyhedron

How can I prove that the point that is maximally distant to a specific point in a convex polyhedron must be in a vertex of the polyhedron?
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44 views

Is this a discrete time Lyapunov function?

I have an algorithm to optimize a process. It is a discrete time algorithm. Every iteration of this algorithm changes the state of the process. I found a function, say $f(s)$, where $s$ is the state ...
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197 views

Minimum of sum of increasing and decreasing function

Suppose we have a function $f(x)$ defined for integer $x$ in some bounded interval, which is positive and increasing $$f(x+1)\geq f(x)\\ f(x)>0$$ , and a function g(x) which is positive and ...
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83 views

Existence of global minimum

Could someone help me with this problem? Let $C$, $D$ convex and closed sets such that the intersection is empty. I want to show that the function $f: \mathbb{R^n} \to \mathbb{R}$ defined by $f(x) = ...
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70 views

Maximize $x^2+y^2+z^2$ on $x^2+y^2+4z^2 = 1$

Hi this is a lagrangian optimization problem. Essentially as the title says, the question is asking us to maximize (if possible) $x^2+y^2+z^2$ on $x^2+y^2+4z^2=1$. I started by the standard ...