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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Optimal control with non-convex cost function

For a controllable linear system $$ \dot{x}=Ax+Bu $$ with state $x=(x_{1},\ x_{2},\ \cdots,x_{n})^{T}$ and the non-convex cost function \begin{equation} \tau(x)= \int_{t=0}^{T} ...
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36 views

Solving Nonlinear System for two variables

I have an optimization like below: $\text{ minimize } \sum_k - w_k\log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ $\hspace{20mm} 0 \leq w_k \leq 1$ I can form the Lagrange ...
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2answers
31 views

Lagrange's multiplier method find the highest and the lowest point

Plane $x+y+z=12$ intersects the paraboloid $z=x^2+y^2$ find the highest and the lowest point of this cross-section. What should i do here? I need help solely when it comes to transforming this ...
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1answer
35 views

Unconstrained Optimization - Minimum Distance Between Point and Curve

Background: I am writing some software can fit a mathematical curve to data using different regression techniques. I currently have Ordinary Least Squares and Least Absolute Deviations mostly ...
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2answers
51 views

Maximizing area under $y=e^{−{∣x∣}}$

The coordinates of the point $M(x,y)$ on $y=e^{−{∣x∣}}$ so that the area formed by the coordinates axes and the tangent at $M$ is greatest is what? I tried to plot the graph but after that I'm not ...
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Best basis selection problem with inequalities solution methods

I'm solving an optimization problem in form $\min \sum x$ subject to, $ A x = b$ (1) $ g x \leq d$ (2) $ x \geq 0$ (3) Optimization variable is x. The number of rows of A is considerably ...
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3 views

Show that a function is quasi-concave by finding a transformation of the function

Assume the function u:R2+ -> R given by u(x1, x2) = (x1+1)(x2+1) Show the u is quasi-concave by finding a transformation of u that is concave Show (without using Hessians) that u is not a concave ...
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1answer
486 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
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49 views

Optimizing over an infinite set of variables

This may be a very basic question, but it's been a while since I did any optimization. Suppose I have a sequence $(x_i)$, $i=1,2,\ldots$ in the $\ell^2$ space and the following optimization problem: ...
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1answer
393 views

Matlab: need help with optimization

I am trying to minimize the objective function over [x(1),x(2)]: exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1)+b subject to constraint ...
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4answers
845 views

The median minimizes the sum of absolute deviations

Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x| $$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take ...
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how to project optimal parameters on to feasible region

Hi: I'm trying to understand the concept of projection and I created a toy example that might help me to do that. Suppose that I have a non-linear optimization with 3 parameters theta_1, theta_2 and ...
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4answers
71 views

Find the minimum value of $f(x)=\sum_{i=1}^n |x-a_i|$. [duplicate]

Let $a_1 \lt a_2 \lt \dots \lt a_n$. Find the minimum value of $f(x)=\sum_{i=1}^n |x-a_i|$. My guess is the minimum occurs at the middle point. However, I don't know how to show this since I can't ...
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Minimizing $\frac{a^TXb}{\|a\|\|b\|}$ given certain constraints.

I'm solving a problem which I have reduced to maximizing $\frac{a^TXb}{\|a\|\|b\|}$ given that $X = (AA^T)^{-1/2}AB^T(BB^T)^{-1/2}$, $a = (AA^T)^{1/2}a'$ and $b = (BB^T)^{1/2}b'$. In this case, I ...
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2answers
576 views

invex functions and their usefulness?

An invex function $f$ is a differentiable function from $\Bbb R^n$ to $\Bbb R$ that for some function $\eta : \Bbb R^n \times \Bbb R^n \to \Bbb R^n$ satisfies for all $x, u$, $f(x) - f(u) \geq \eta(x, ...
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1answer
20 views

Find diffeomorphism transforming the following areas:

Find diffeomorphism transforming the following: interior of the triangle T with vertices in $(0,0),(0,1),(1,0)$ onto the interior of the circle of radius 1 and centre in $(0,0)$. Obviously i am ...
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1answer
20 views

Max volume of a cuboid given constraint

Find maximum volume of a cuboid for which the sum of three dimensions does not exceed $108$. I think expression to maximize is: $\left( 108-y-z\right)yz$. From partial derivatives I got that max will ...
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1answer
14 views

Max volume of a cylinder

Find maximum volume of a cylinder of which the sum of height and the circumference of the base does not exceed 108 cm. How to solve this? Precisely what is the expression that should be minimized? How ...
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28 views

Dynamic Optimization - Transversality Condition for Infinite Horizon Case

When solving dynamic optimization problem such as $$ \max \int_0^\infty f(t,x,x')dt $$ $$ \ s.t. x(t_0)=x_0 $$ we can use the Euler equation to obtain a differential equation to solve. ...
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1answer
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Find a point from the area tha is closest to the other point.

Given: $$A=\{\left(x,y,z \right)\in \mathbb{R}^3 : 2x-3y+z=1 \}$$ Find a point $\left(x,y,z\right)\in A$ that is closest to $\left(3,-2,1 \right)$. I do not know how to solve that problem, i need a ...
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variable transformation in optimization

I have an optimization problem with two sets of parameters, $x_i \in [0,1]$ and $y_k \in [-\frac{\pi}{2},\frac{\pi}{2}]$ where $i,k \in \{1...n\}$ are indices. One way to solve this problem is using ...
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3answers
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Finding the distance from the origin to the surface $xy^2 z^4 = 32$ using the method of Lagrange Multipliers

Problem: Find the distance from the origin to the surface $xy^2z^4 = 32$. Attempt: The Lagrange equation for this problem is $L(x,y,z, \lambda) = x^2 + y^2 + z^2 + \lambda (xy^2 z^4 - 32)$. Setting ...
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Calculating amount of cubes that fit in a sphere

I know that the problem of finding out how many spheres can fit in a cube is a commonly asked and well documented one, but I am struggling to find anything on the inverse of the problem, namely: ...
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2answers
529 views

linear least squares minimizing distance from points to rays - is it possible?

I'm writing a tool whose purpose is to process data from a sensor that provides the true bearing to a target, and combine measurements taken at various times into an estimate of the target's position ...
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1answer
78 views

Calculate the maximum area (maximum value)

TX farmer has 100 metres of fencing to use to make a rectangular enclosure for sheep as shown. He will use existing walls for two sides of the enclosure and leave an opening of 2 metres for a gate. ...
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3answers
1k views

Where to build a bridge to cross a river in the shape of an annulus

There is a river in the shape of an annulus. Outside the annulus there is town "A" and inside there is town "B". One must build a bridge towards the center of the annulus such that the path from A ...
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59 views

The maximum of a functional

Is the following statement true or false? $$ \max F \left( \theta \right)= \int_{\rho_{min}}^{\rho_{max}} g \left( \rho \right)\pi\left(\rho,\widehat{\theta \left( \rho \right)} \right)d\rho$$ ...
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30 views

Minimizing the sum of the $4^\text{th}$ power of a matrix entries.

Consider a real $n\times n$ matrix $X$. Suppose I would like to minimize the sum of the squares of its entries as a penalty term in some convex minimization. I can write the term using the Frobenius ...
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302 views

Intuition behind accelerated first-order methods

$\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ Suppose that we want to solve the following convex optimization problem: $\min_{x \in \mathbb{R}^n} g(x) + ...
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Show that inf $f(x)$ is achieved. Find $\inf f(x)$.

Let $$\Sigma = \{x\in R^3: x_1x_2 +x_1x_3 +x_2x_3=1 \}$$ and $$f(x) = x_1^2 + x_2^2 + \frac{9}{2} x_3^2$$ a) Show that $\Sigma$ is a smooth surface in $R^3$. b) Show that $\inf_{x\in\Sigma}$ f(x) ...
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1answer
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Finding KKT conditions for nonlinear optimization problem.

I have an optimization like below: $\text{ minimize } \sum_k - \log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ I can form the Lagrange of the problem as below: $L(x, ...
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Scaling issue with Gradient descent methods

As is the common knowledge that gradient methods are affected by scaling issue of the variables. For example, If minimizing a function of say 2 variables $x_1$, $x_2$. Both variables have different ...
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1answer
459 views

How to see that K-means objective is convex?

I'm trying to proof that the objective of the K-means clustering algorithm is non-convex. The objective is given as $J(U,Z) = \|X-UZ\|_F^2$, with $X \in\mathbb{R}^{m\times n}, U\in ...
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1answer
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Conjugacy relation in the primal and dual problem

The following is my derivation in the Conjugacy relation in the primal and dual problem. I am shaky in it; so hope for some advices. Consider the following problem, $f(x),g(x)$ are convex ...
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Some problems in finding conjugate function

Ask the following fundamental problems: How to derive the conjugate function of $g(y)$ if given "$\underset{y \geq 0}{\text{sup}}\{g(y)-y^Tx\}$"? My attempt is as following: \begin{align*} ...
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Maximizing the area of a transformed rectangle within some bounds

I need to solve this problem for a program I'm writing, but I'm struggling a bit with the maths behind it. Given a rectangle $R_{max\_layout}$ and a $3\times 3$ transformation matrix $M_{transform}$, ...
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2answers
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The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.

The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.The area of the triangle will be maximum if the angle between them is: ...
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Difference between maximum and minimum?

If I have a problem such as this: We need to enclose a field with a fence. We have 500m of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the ...
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1answer
43 views

Pontryagin's Maximum Principle as a sufficient condition?

It is know that Pontryagin's maximum principle provides in general only a necessary condition in the following sense: The ODE system which is known to be solved by the optimal control may have ...
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22 views

Minimum vertex cover of two edge disjoint perfect graphs

How well can the minimum vertex cover of the union of a perfect graph and bipartite graph (the two graphs are edge disjoint but not vertex disjoint) be approximated?
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Integer (Binary?) Optimisation of a problem

got a question regarding maximal optimisation of a problem. Refer to the table below: $$\begin{array}{c|c|c|} & \text{Area A} & \text{Area B} & \text{Area C} & \text{Area D} \\ ...
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Solving linear constrained optimization problem

So I have the following constrained optimization problem to optimize a circuit (electrical engineering) that I am working on: Minimize the following expression (power dissipation): $$I_{B1}(V - C_1) ...
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Find the critical curves for the following functional with subsidiary conditions

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1}\left(y'^2+z'^2-xyz'-yz\right)dx$$ with subsidiary conditions : $$\int_{0}^{1}\left(y'^2-xy'-z'^2\right)dx=2$$ ...
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Solving Nonlinear system with logarithmic objective function

I have my objective function as : $\hspace{25mm} \text{Minimize} \sum_k- \alpha_k \log_2 W_k$ $\hspace{25mm} \text{subject to}: 0\leq W_k \leq q', 0 \leq \alpha_k \leq 1 $ $\hspace{25mm} k ...
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29 views

Placing Circles Onto Lines For Optimality

Suppose you have a yet to be determined number of vertical lines with length 50 on which you'd like to place as many circles as you can. Each circle is 10 units in diameter and its outside edge must ...
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Winning the relay race for your team

Relay race, members of a team of three take turns running from the point P to a point on the circle; To A for the first, B for the second, and C for the third, starting and returning to point P, ...
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64 views

Given matrices $B$ and $C$. What is the value of $L$ that minimizes the value $||L^T \times B \times L - C||_F$?

Where $L \in R^{m \times n}$ and $B \in R^{m \times m}$ and $C \in R^{n \times n}$ $B$ and $C$ are symmetric positive semi-definite. Where $\times$ denotes matrix multiplication and $||.||_F$ ...
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Minimum of a function given by integral and inequality type constraint

I need your help with the following problem I want to minimize $$2a + \int_0^1 tx(t) \, dt \to \min$$ s.t. $$1 - a - \int_t^1 x(s) \, ds \leq 0\text{ a.e. }t \in (0,1)$$ $$x(t) \geq 0 \text{ a.e. ...
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1answer
51 views

How can I solve a nonlinear optimization problem where constraint contains exponential term?

I have the optimization problem as below: $\hspace{10mm} \text{Maximize} \sum_{k} \alpha_k {R}_k $ $ \hspace{10mm}\text{subjcet to:} $ $ \hspace{10mm}\exp \left[ - (2^{{R}_k } -1) \left( ...
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Soviet Optimization books

I am aware of an answer on Soviet math books here: Soviet Russian Mathematical Books and the book by Boris Polyak on non linear optimization. I am also aware of a few books by Kantorovich which I do ...