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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Shortest smooth paper Möbius Strip

I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What ...
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Derivatives - optimization (minimum of a function)

For which points of $x^2 + y^2 = 25$ the sum of the distances to $(2, 0)$ and $(-2, 0)$ is minimum? Initially, I did $d = \sqrt{(x-2)^2 + y^2} + \sqrt{(x+2)^2 + y^2}$, and, by replacing $y^2 = 25 - ...
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Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. Find the maximum value of $2x - 5y$.

Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. Find the maximum value of $2x - 5y$. I do know how to solve this problem using trigonometry, however I need to solve it by using vectors. ...
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36 views

Weighted least squares with nuclear norm minimizaiton, how to optimize?

Nuclear norm minimization is very popularization and formulation is least squares term with nuclear norm term as following, $$\min\limits_{X} ...
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9 views

Lipschitz Number in Gradient Descent

During gradient descent, if an objective function's value is greater than the previous iteration, would use of an orthogonal vector to the update vector be advantageous? Regarding trust regions, the ...
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477 views

Graph theory problem (edge-disjoint matchings)

Find the smallest number $x$ so that if an $n$-vertex simple graph has at least $x$ edges then it contains $k$ pairwise edge-disjoint perfect matchings* ($k$ is a positive integer, $n$ is an even ...
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2answers
22 views

optimization for the area of a garden

I have been working this problem for awhile and cannot seem to solve it even though its probably easier than I think... There is a rectangular garden that needs fencing. For one side of the fencing ...
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1answer
38 views

Find Min: $A= \frac{bc}{a(b+2c)} +2 [ \frac{ac}{b(c+a)} + \frac {ab}{c(2a+b)}]$

Given $a,b,c>0$ such that: $ \frac{4a}{b} (1+ \frac{2c}{b}) + \frac{b}{a} (1+ \frac{c}{a})=6$ Find Min: $A= \frac{bc}{a(b+2c)} +2 [ \frac{ac}{b(c+a)} + \frac {ab}{c(2a+b)}]$ My try: Let: ...
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570 views

Gradient descent with inequality constraints

Suppose we are given a convex function $f(\cdot)$ on $[0,1]$. One wants to solve the following optimization problem: \begin{equation} \begin{aligned} & \text{minimize} && \sum_{i=1}^n ...
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Gluing two strongly convex function

Definition: We call $f:\mathbb{R}^n\rightarrow \mathbb{R}$ a $\lambda$-strongly convex function iff for every $x,y\in \mathbb{R}^n$ and $t\in[0,1]$ it follows $$f(tx+(1−t)y)\leq ...
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Present and future value: selling now vs selling for a higher price later

A wine dealer contemplates whether to sell his bottle of wine for $\$30$ today, or wait to sell it in the future. If he sells it in the future, then he can sell it for a higher price. However, the ...
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Complexity of finding set of sets with maximum cardinality and constrained coverage.

Given a set of sets $S = \{S_1, S_2, \dots, S_n$}, let $S^{'} \subset S$ be the largest subset of S that obeys $\left| \bigcup_{S_i \in S^{'}}{S_i} \right| \leq k$. What is the complexity of finding ...
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17 views

Do quasi-Newton methods check the second-order optimality condition?

I have a practical question about quasi-Newton methods. In quasi-Newton methods, Hessian matrix is approximated. It seems to be impossible for them to check the second optimality condition. In ...
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14 views

Reference Request for Penalty Method for Optimal Control?

Is there a good book or review article to read about the methods like penalty method, method of duality and method of relaxation in problems of calculus of variations and their relations to optimal ...
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398 views

Absolute extrema

Find the absolute extrema of the function $f(x,y)=2xy-x-y$ over the region of the xy-plane bounded by the parabola $y=x^2$ and the line $y=4$ I was wondering if I needed to use Lagrange multipliers to ...
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representing an iteration loop in math

I have a computation step where $$ a_{n+1} = f_1(a_n) $$ That is, $a$ at step $n+1$ is some function $f_1$ of $a$ at step $n$. I want to iterate till an $N$ where $f_2(a_N) = b$ (where $f_2$ is ...
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34 views

meaning of Farkas' Lemma

Quoting from Jorge Nocedal's Numerical Optimization second edition, page 326 bottom to page 327, Farkas' Lemma Let the cone K be defined as in (12.45). Given any vector $g \in \mathbb{R}^n$, we ...
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connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian

In the context of solving linear programs, the big-M method refers to adding additional variables to the problem such that there is, as far as I understand it, a trivial basic feasible solution. In ...
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12 views

Minimum of a cubic fitted to two points and their derivatives

I'm trying to understand a line search method used to find a step length in a minimsation algorithm. There is an interval $[a, b]$ containing desirable step lengths and there are two previous ...
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22 views

How to find optimized value of two variables

I have two variables: $\kappa_y$ and $\kappa_x$ And three functions: $M_y$=$M_y$($\kappa_y$, $\kappa_x$) $M_x$=$M_x$($\kappa_y$, $\kappa_x$) $F_z$=$F_z$($\kappa_y$, $\kappa_x$) All these three ...
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2answers
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Why a convex cone cannot have more than one extreme point?

The way I define an extreme point is : A point which cannot be defined as a convex combination of two distinct points. I'm not able to extend this and show why a convex cone cannot have more than ...
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simplex algorithm - minimization

So I get the basic concept of simplex algorithm but I am working on a project where I have to implement any linear programming algorithm (I chose simplex method) to minimize a function, but I don't ...
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40 views

minimize expression

How can the following expression be minimized wrt w: $$ \frac{w^T D w}{w^T S w}, $$ where $w \in \mathbf{R}^n$, $D \in \mathbf{R}^{n \times n}$ symmetric, and $S \in \mathbf{R}^{n \times n}$ ...
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Is there a convergence proof for ADMM applied to biconvex/bilinear problems?

I wonder if there is a local convergence proof for ADMM applied to biconvex problems? More specifically, my problem is as follows: $\text{minimize}_{x,y} f(x) + g(y) + \| y \circ Ax \|_2^2 $ , ...
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Why Was Backprop Invented?

I'm currently researching artificial neural networks and I keep wondering why do we use "backpropagation" to train a neural network. An ANN is basically just a very large and complex function ...
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22 views

convergence of steepest gradient descent

The description of gradient descent in Wikipedia says: $$x_{n+1} = x_n - \gamma_n\nabla F(x_n)$$ for $n = 0,1,2,...$ Suppose that $x_n$ converges to $x$. Then, is it always true that $\nabla F(x) = ...
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Direction in Dual Simplex method

In the dual simplex problem, when primal become inconsistent then dual have direction. How can we find this direction using dual simplex algo ?
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Empirical likelihood method to Find the order of a parameter given a set of constraints

Firstly we assume that $X_1,...X_n$ are order statistics($X_i\leq X_{i+1}$) from an i.i.d sample of random variables and let $r$ be integer and $r\geq1$. Start with the Equation (1) as following ...
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Maximize $x^tQx$ s.t $||x|| \leq 1$

I would like to verify that I have solve correctly the following problem: $maximize\; x^tQx\; s.t. ||x|| \leq 1$ Using Lagrangian multipliers I derived the following: $2Qx = λx$ So $x$ are the ...
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Optimal strategy in the following game:

In this game, 12 hidden D6s are rolled and summed. The player is given the total of the rolled dice. The player will then guess a number from 1 to 6. If there is a unrevealed dice with that number, ...
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Lagrange Multiplier for clustering with size constrains

I'm trying to solve a clustering problem with size constrains. Minimize $J=\sum_{i=1}^c\sum_{j=1}^n {{u_i}_j}^2{d_i}_j$ Subject to $\forall 1\le j\le n : \sum_{i=1}^c {{u_i}_j}=1$ and ...
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(Empirical likelihood method) Find the order of a parameter given a set of constraints

Firstly we assume that $X_1,...X_n$ are order statistics($X_i\leq X_{i+1}$) from an i.i.d sample of random variables and let $r$ be integer and $r\geq1$. Start with the equation (1) \begin{equation} ...
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1answer
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How to find the largest disk in a square when there are points we must avoid?

We have $n$ points $X =\{x_1, x_2, \dots, x_n\}$ inside (let's say) the unit square $Q$. We must find a disk $D\subset Q$ such that none of the points of $X$ are inside the disk. (The points can be on ...
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69 views

T-shaped polygons

Is there any coefficient that can indicate T-shaped polygons ? Examples of T-shaped polygons:
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Lagrange multiplier method

Question 1: Could somebody please refer me to an introduction to Lagrange multipliers which is easy to read but still in full generality? Question 2: I am interested in particular in the following ...
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Linear programming right hand side is abstract

I am solving a problem in network flows which uses linear programming to find the minimal point of a. the problem stated is to minimize a with contain to: $x_1 + x_2 \le a$ $x_3 + x_4 \le a$ ...
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Using semidefinite programming to solve the following problem

I am struggling with the following problem, and wonder is SDP can help: $$\mathrm{maximize\ } \alpha_{10}+\alpha_5+(\alpha_2+\alpha_8)/2 \mathrm{\ subjected\ to\ } \mathrm{T_1}\succeq0, ...
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How can I compute fast the minimum of a linear plus Kulback-Leibler on the unit simplex?

Given $a, x^0 \in \mathbb{R}^n$ I wish to compute $$\min_{x \in \Delta_n} a^t x + \sum_{i=1}^n x_i\log(x_i/x^0_i) - x_i +x^0_i $$ where $\Delta_n$ is the unit simplex $\{x \in \mathbb{R}^n \mid ...
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How find this minimum

Help me! Let $x,y,z\ge0$ such that: $xy+yz+zx=1$. Find the minimum value of: $A=\dfrac{1}{x^2+y^2}+\dfrac{1}{y^2+z^2}+\dfrac{1}{z^2+x^2}+\dfrac{5}{2}(x+1)(y+1)(z+1)$ I found minimum value of $A$ ...
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29 views

Minimizing Unintegrable Exponential Function

I am trying to develop an algorithm which minimizes an unintegrable function. I don't have a strong mathematics background and am unaware of such strategies. My integral is of the following form: ...
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How does one minimize determinant on this linear system?

I have a problem where want to determine loading factors such that can reduce a covariance matrix as close to 0 volume as possible, where the original covariance matrix is specified as: $\ \ \ \ \ \ ...
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Maximize linear combination

Given a set of positive values $v_1, v_2, ..., v_n$ and a set positive of factors $\alpha_1, \alpha_2, ..., \alpha_n$, both ordered increasingly, show the maximum linear combination you can get is ...
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Formulation of a SDP problem in a standard form with constraints on blocks

I have a semidefinite problem, \begin{align} \min_{\beta,\eta,R} &t\notag\\ \textrm{s.t.} & \beta ,\eta \geq 0\notag\\ &\Bigl[\begin{matrix} K\odot R&&1+\eta-\beta\\ ...
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Optimising money in bonds

Hi im doing an optimisation problem but dont understand what the terms mean. Suppose someone wants to invest $\$110,000$. They have $4$ choices as to what they invest their money into: $\bullet$ ...
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Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
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Does a larger change in $R^2$ indicate a given parameter is “more important” in model fitting?

I have a set of experimental data, and a given model which is supposed to fit that data. Let's say this model has $n$ parameters. I need to write a parameter estimation algorithm to determine the ...
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Is optimal solution to dual not unique if optimal solution to the primal is degenerate?

If optimal solution to the primal is degenerate, does it necessarily follow that optimal solution to dual not unique? That is, is uniqueness an unnecessary assumption? Spin-off from here. In my ...
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Book on duallity and sensitivity in nonlinear optimization

I am looking for a recommended book on duallity and sensitivity in nonlinear optimization, as duallity and sensitivity is a well studied topic in LP , I am struggeling to find books in this subject ...
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Continuity of optimisation problem

Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$ I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and ...
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Convergence rate - Convex optimization

What is the best known algorithm in terms of convergence rate for unconstrained convex optimization and under what assumptions? $\min_{x} f(x)$ where $f(x)$ is a given twice differentiable convex ...