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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Optimization in Calculus

As you can see I found the equation but I don't know how to find the points. As far as I tried was $(7, 49)$ but it was wrong.
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Random Rotation of Points using Householder matrices

I have $N$ points in $D$ dimensions, were $D$ is big, for sure more than $100$. $N$ is also big. The goal is to produce an algorithm in my code, that will take as input this dataset and will give ...
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1answer
32 views

How to find all stationary points of $ \alpha\|v\|^2-\|x^Tv\|^2+\|g^Tv\|^2$

Let $v,x,g$ be three vectors and $\alpha$ be a constant. The problem is $$\min\limits_v \{\alpha\|v\|^2-\|x^Tv\|^2+\|g^Tv\|^2\}$$ where $\|v\|^2=\sum\limits_{i=1}^{|v|}v_i^2$ and $|v|$ is the ...
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Proper name for the problem (finding optimal discrete function)

Given a set $D = \{d_1, d_2, ..., d_N\}$, a set of some subsets of $D$, $D^\ast$ and a set of classes, $C = \{c_1, c_2, ..., c_M\}$, I want to find function, that maps a sequence $({d_i}_1^\ast, ...
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16 views

Dependency of the Lagrange multipliers

Let $F, g$ be a polynomias in $n$ variables and consider the optimization task $\min F(x) s.t. g(x)=0$ . In Order to solve this with the Lagrange method, one has to find a multipliers. Can one say ...
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6 views

Objective function reaches a plateau

I have an objective function and I am trying to minimize it. I noticed that if the objective function is far away from the "solution" it decreases (convergence). Once I start close to the solution the ...
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30 views

Minimizing a multivariable function in several variables

I would like to show that a certain function is negative, to help establish asymptotic stability via a Lyapunov function for a system of differential equations. This is exactly what I need help on: ...
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12 views

Estimating objective function weights from Pareto front

Let's say I have 2 functions $f_1(x)$ and $f_2(x)$. I ran a multi-objective optimization method to obtain the Pareto Front. Now, take any point P on the Pareto front. Assuming the Pareto Front to be ...
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1answer
19 views

Rank degenerate non negative least squares

I'm following an algorithm in the book "Solving Least Squares Problems" by Lawson and Hanson (#15 in Siam's Classics in Applied Mathematics) for solving non negative least squares. That is, minimize ...
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0answers
51 views

The dual simplex algorithm

Following is the dual simplex algorithm, adapted from p. 283 of Daniel Solow's "Linear Programming, An Introduction to Finite Improvement Algorithms", Elsevier Science Publishing Co., Inc., 1984. I ...
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22 views

Linear assignment problem - would this constraint also be correct?

In the linear assignment problem, could the constraint $\sum_i x_{ij} = 1$ also be replaced by $\sum_j \sum_i x_{ij} = n$? Because I think this also ensures that all tasks are executed, and the ...
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1answer
50 views

How can I modelize a weekly menu and minimize the total number of ingredients it contains

Hi and many thanks for reading this question. I want to create an algorithm that will minimize the total number of ingredients that are in a weekly menu. A menu is made of several recipes, for ...
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22 views

Primal-dual subgradient method

In these notes, an extension of the subgradient method is presented in Section 8 (page 30). The method is described so quickly and neither convergence analysis (compared to classical subgradient for ...
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14 views

Understand and an algorythm to Maximize number of triangles from a set of points on XY plane

Given: Set of points (x, y) Looking to: Maximize count of triangles that can be formed. Each triangle which is enclosed within another (with/without shared edge) will be counted again. Specifics on ...
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15 views

Threshold in a maximisation problem (with KKT conditions)

I'm looking to maximise with respect to $x_i$ $$L = \sum_{i= 1}^n y_i \frac {x_i^{1 - \epsilon}}{1 - \epsilon}$$ subject to $ \sum_{i= 1}^n x_i = B $ and $x_i \ge 0$ for all $i$, where $y_i$, $B$ ...
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2answers
53 views

Optimizing elementary symmetric polynomial on the unit sphere

I'd like to optimize $x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4$ on the unit 4-sphere. I'm thinking I should do lagrangian optimization, but I'm having trouble solving the resulting ...
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2answers
72 views

When are there no critical points?

Is there ever a time when there are no critical points of a function? For example, I am trying to find the critical points and the extrema of $\displaystyle f(x)= \frac{x}{x-3}$ in $[4,7]$ I am not ...
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1answer
176 views

Characterization of sphere.

I'm editing the question because I think the previous formulation was leaving a key element of the problem out and that was making it impossible to answer the question. I tried to update/improve the ...
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2answers
75 views

Maximizing discrete probability

I'm stuck with the following problem: Let's assume we have two buckets: bucket one contains $k$ white spheres and $l$ red spheres. Bucket two contains $n-k$ white spheres and $n-l$ red spheres (n a ...
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2answers
25 views

optimization problem: finding an hyperplane separating one point from a set of pointy maximizing the distance

I have this problem: I have a set of n-dimensional points $P$. I have one more n-dimensional point $q$. The points in $P$ are linearly separable from $q$ (i.e. it always exists an hyperplane $n^t x ...
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2answers
66 views

Forbidden range for a linear programming variable

I would like to express a linear program having a variable that can only be greater or equal than a constant $c$ or equal to $0$. The range $]0; c[$ being unallowed. Do you know a way to express this ...
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1answer
56 views

minimize distance

consider a two dimensional system. two points are given whose co-ordinates are $(h1,h2)$ and $(k1,k2)$. I want to minimize the distance between these two points with the condition that person has to ...
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2answers
52 views

Minimize Sum a_i / Sum b_i over subsets

I have two positive finite sequences $a_i$ and $b_i$, with $0 \leqslant i \leqslant n$. The problem is to find the subset $I$ of $\{0, ..., n\}$ that minimizes: $$\frac{\sum_{i \in I} a_i}{\sum_{i ...
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51 views

Linear Programming with percentage constraints

I need some guidance with this Linear Programming problem, I want to maximize profits subject to different constraints, the problem is Maximize Goods subject to Min and Max Percentage Volume and ...
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2answers
74 views

Formulation and computation of “the” unique median of an even-sized list

Consider an even-sized set of numbers $X = \{x_k\}$, such as $X = \{1, 2, 7, 10\}$. The median $m$ is defined as: $$m = \mathrm{arg \min_x} \sum_k \lvert x_k - x\rvert^1$$ Any $m \in [2, 7]$ is a ...
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1answer
68 views

Division of plane into equal area regions

We divide a plane ($\mathbb{R}^2$) into infinite number of regions each of area equal $1$. We can use only (one-dimensional) curves which may meet at points. Fix a point $p$ on a plane and consider ...
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2answers
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Is this condition sufficient to ensure the locally convexity of a function at a given point?

Given $\bar x\in \mathbb R^n$. Let $f:\; \mathbb R^n\to \mathbb R$ be a nonconvex continuous function on $\mathbb R^n$ satisfying the followings (i) $f$ is not differentiable at $\bar x$, (ii) There ...
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Proof of sufficient condition of existence of Lagrange multipliers

Consider the optimization problem $$ (P) \quad \inf\{ f(x) : g_i(x) \le 0, i = 1,\ldots, m, x \in \mathbb R^m \} $$ where $f, g_i : \mathbb R^n \to (-\infty, \infty]$ are convex and $0 \ne ...
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1answer
32 views

Sufficient condition on convex function such that $f(x) > -\infty$ for all $x$.

Let $f : \mathbb R^n \to [-\infty, \infty]$ convex and let $f(\overline x) > -\infty$ for $\overline x \in \mbox{int}(\mbox{dom}(f)$. Show that $f(x) > -\infty$ for all $x \in \mathbb R$. ...
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Solve for transform of rotating frame to fixed frame given points in rotating frame and a planar constraint

Say there are 2 coordinate systems, with one orbiting around the other. Call one fixed ƒ and the other rotating ρ. The goal is to find the transform between the two frames. What's known is A set of ...
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1answer
25 views

Projection theorem for compact differentiable manifold

By Hilbert projection theorem, if $x\in\mathbb{R}^n$ and $D$ is a closed subset of $\mathbb{R}^n$ then the optimization problem $$\underset{y}{\min} \|x-y\| \ s.t. \ y \in D \quad\quad (P1)$$ has an ...
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1answer
44 views

binomial coefficient: maximum value

For $n\rightarrow \infty$ we consider $$f(p)=\sum_{j=c}^n {n\choose j} p^j (1-p)^{n-j}.$$ We are interested in $\hat{p}:=\arg \max_p f(p)$. Can we say something about $\hat{p}$ dependent on $n$ and ...
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How find this maximum of the $\sin^2{\theta_{1}}+\sin^2{\theta_{2}}+\cdots+\sin^2{\theta_{n}}$

Question: let $\theta_{1},\theta_{2},\cdots,\theta_{n}\ge 0$,and such $$\theta_{1}+\theta_{2}+\theta_{3}+\cdots+\theta_{n}=\pi$$ find the $P$ the maximum of value $P(n)$ ...
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1answer
36 views

Binomial Coefficient: monotonically decreasing in this range?

relating to this question, I'd like to ask a further one. Again we have $$f(x)={k-1 \choose x-1} p^x (1-p)^{k-x}$$ We know that this term is maximal for $x=kp$, before increasing, afterwards ...
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1answer
23 views

monotonicity of binomial coefficient

I am interested in $$f(x):={k-1 \choose x-1} p^{x} (1-p)^{k-x}.$$ How do I find out in which Domain this function is monotonically increasing, in which it is monotonically decreasing? For which $x$ ...
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1answer
31 views

Optimization with both equality and inequality constraints

I need to minimize the following quantity: $$\min x_1^{-1/n}- \left(1-x_2 \right)^{-1/n}$$ subject to: $1-x_1-x_2=\gamma$ and $0<x_1+x_2<1$ $\gamma$ being a constant. Had it been two ...
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1answer
42 views

Elegant way to solve this extreme value problem

I want to show that $$ \sup_{(x,y)\in \mathbb{R}^2 \setminus \lbrace (0,0) \rbrace} \frac{(ax+by)^2}{x^2+y^2} =a^2+b^2 $$ where $a,b \in \mathbb{R}$ are fixed (this problem appears when one tries to ...
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Non-convex maxmin optimization

I am dealing with the following maxmin optimization problem: $c^*, x^* = \arg\max\limits_{c \in C, x \in X} [f(c, x) + \min\limits_{\tilde{x} \in X} g(c, \tilde{x})] $ $f$ and $g$ are differentiable ...
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23 views

How to characterise this non-linear optimisation (linear objective function, non-linear constraints)

I was wondering if someone may be able to help me characterise this optimisation problem as I am struggling to find a numerical library that will solve it and I suspect it is because I am using the ...
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2answers
46 views

Extreme value problem, maximize ratio of volume to surface area

For a cylindrical can, how to choose the ratio of the height to the radius such that the ratio of the volume to the surface area gets maximized? The volume ist $V = \pi r^2 h$ and the surface ...
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closest points on two Line Segments

I am looking for a general formulation to find the closest points on two line segments: What I was thinking about is to define our lines as: $ P1 + s * (P2-P1)$ $ Q1 + t * (Q2-Q1)$ Where $P1 , P2, ...
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what does “modular” mean?

I find some similarity of the concept "modular set functions" to the cardinality function. But I don't see the cardinality function is also called "modular" or something else. I wonder what "modular" ...
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How to find iso function value points without exploring all points in 2D space

Consider a 2D graph with dim1 and dim2 represented as X and Y respectively. The range of X and Y are 1 to 100. Hence there are 10000 points in the 2D space. Each point in the space is some function of ...
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27 views

Minimize function with constraint

I have a Markowitz problem : Min $x^T*C*x$ $x : {x_1 , x_2 ... x_n}$ is a vector of size $N$ $C$ is a known matrix $[N \times N]$ 1) $∑ x_i$ = 1 2) $x_1 < 0 $ I can minimize the function ...
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1answer
27 views

The convexity of convex function's range

Given a convex function $f\colon X \to \mathbb R$ with convex domain $X \subseteq \mathbb R^n$, is the range of $f$ a convex set also?
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What does “modular” in “modulr functions” mean?

From Wikipedia If $\Omega$ is a set, a submodular function is a set function $f:2^{\Omega}\rightarrow \mathbb{R}$, where $2^\Omega$ denotes the power set of $\Omega$, which satisfies one of the ...
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Generate primitive integer triangles

I am working on a program where I need to generate all primitive integer sided triangles with lengths $x,y,z$ such that $1\leq x \leq y \leq z$ and $x+y \leq m$. A primitive integer sided triange is ...
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Convergence of projected gradient method for non-convex functions.

Is there a proof of convergence for the projected gradient method for non-convex functions? By projected gradient method I mean the following (shortened) algorithm for $f: U \rightarrow \mathbb{R}$ ...
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14 views

Restating optimization problem for quadratic programming

I'm working on implementing an author disambiguation algorithm as described in Torvik et al's paper. I've got most steps done, but am completely stumped on implementing a quadratic optimization step. ...
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1answer
24 views

Hessian of non-differentiable function

Given a function $f = \max\{f_1,f_2\}$ with $f_1,f_2$ convex and differentiable, I know I can calculate the subgradient of $f$. Is there also an equivalent of the subgradient for the (sub)Hessian? ...