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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How can we maximize the following functional?

$\max_{} \; -\frac{1}{6} \lambda_1^2 + \lambda_1 + \int_0^1\left( \lambda_1 \lambda_2(t) (1-t) - 0.5 \lambda_2^2(t)- 2.5 \lambda_2(t)\right) dt$ s.t $\lambda_1\geq0$, and $\lambda_2(t) \geq 0$ for ...
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Trace minimization-Revised

The problem is as follows: $\displaystyle\min_{V}$ trace($V^TH^T\Phi HV$)$\\$ s.t. $V^TV=I_d$ in the case when $H$ is not known. When $H$ is known, the solution is given by the eigenvectors ...
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Find the edges of a polyhedron P.

Given the polyhedron $P = \{v \in \mathbb R^2 \mid Av \le b\}$ with $A = \begin{bmatrix} -1 & -1 \\ 2 & -1 \\ -1 & 2 \\ 1 & 2 \end{bmatrix}$ and $b = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 2 ...
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17 views

Solving an optimization problem with KKT-conditions

I've been studying about KKT-conditions and now I would like to test them in a generated example. My task is to solve the following problem: $$\text{minimize}:\;\;f(x,y)=z=x^2+y^2$$ ...
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Say optimal solution to the primal is degenerate. Does it hold that optimal solution to dual not unique?

I think it's supposed to be that existence of a degenerate and unique solution of the primal implies multiple solutions to the dual, according to this book (pages 141-145, proof of Theorem 4.5). In ...
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10 views

Sufficient condition for constrained extrema

When study Lagrange multiflier theorem. I try to get a sufficient conditions for constrained extremum due to the statement of Lagrange multiflier theorem, that is Given p+1 countinously ...
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1answer
28 views

$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$ for all positive definite $A\in\mathbb{R}^{n\times n}$

Let $A\in\mathbb{R}^{n\times n}$ be positive definite and $v\in\mathbb{R}^n$. Let $\left\|\cdot\right\|_2$ be the Euclidean norm. Can we prove $$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$$ for ...
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22 views

Find maximum and minimum of funсtion on set

I have the task: find maximum an minimum of $$f(x) = x_1(\pi - x_1)\sin x_2 + x_2 \cos x_1$$ on X where $$X = \{x\in R^2\ |\ x_1\in [0, \pi], x_2 \ge 0\}.$$ First thing i did was system : ...
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40 views

Definition functions, integrals on $\mathbb R^{|N|}, \mathbb R^{\mathbb R}$

Is there a standard/reasonable way of defining functions on the sets $\mathbb R^{|\mathbb N|}, \mathbb R^{\mathbb R} $. How about defining integrals over these sets? I guess a function on $\mathbb ...
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calculus optimization: solution two equations

I am to do this maximization problem. I simplified it down to this triangle. I am given no dimensions, instead I am given that G is fixed and H is fixed. The rest are variables. We are to maximize ...
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Quadratic program with complementarity/modular constraints is NP?

Is the following program NP/NP-hard? Any neat way to prove it, or a helpful reference? $\min x^TMx$, subject to $\|x\|_1=1,e^Tx=0$ Here $M$ is a real, symmetric and semidefinite positive matrix, ...
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21 views

calculus optimization problem: give answer in equations

I have a weird question as part of a homework assignment I am having trouble with. I think I am getting tripped up using only letters and no numbers. The question goes: Fred and Sally are adding a ...
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26 views

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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1answer
17 views

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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15 views

How to do optimization

My teacher gave me a very complicated explanation on how to solve an optimization problem so I just wanted clarification. To do so I have laid out what I think is the simplest way to solve it. Take ...
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1answer
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ADMM formalization

I found lots of examples of ADMM formalization of equality constraint problems (all with single constraint). I am wondering how to generalize it for multiple constraints with mix of equality and ...
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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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60 views

An easy question about NP-hard

Consider an optimization problem includes two variables. If we fix the value of one variable, then the optimization problem over the other variable is NP-hard. Can it be concluded that the original ...
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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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30 views

Maximum of norm

Given a matrix $A$ with $N$ rows and $d$ columns, I would like to prove (or disprove) the following. Let $q(f)=\|(\begin{pmatrix} ...
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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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67 views

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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36 views

Effect on Minimizer of Tightening Constraints

The Statement of the Problem: Consider the minimization problem $f(x,y)=14x+20y$ under the constraints $x+2y \ge 4 $, $7x+6y \ge 20$, and $x,y \ge 0$. Don't use the simplex method! (i) Draw the ...
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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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34 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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28 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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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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1answer
17 views

Grouping constrained optimization

I am looking for an efficient solution to solve the following problem. Can anybody help? S is a finite set of elements $k_i$ V is a subset of S, e.g. $v_4$={$k_1$,$k_3$} E is a finite ensemble of ...
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23 views

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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9 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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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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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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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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38 views

Minimizing Norm

I have below problem: Find $\bf C$ to minimize $\|\mathbf A-\mathbf B\mathbf C\|_F$. Given ${\bf B} \in \mathbb R^{m \times n}$, ${\bf B}$ has lin. ind. col. A satisfies: ${\bf DA} = {\bf E}$ , ...
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42 views

How to simplify the summation of log

I have a summation that involve log. I don't know how to solve this summation. I want to find an expression (even a good approximation is enough) for this summation. $\sum_{k=0}^{n}{log(a_k)}$ ...
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49 views

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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Frank Wolfe algorithm in matlab [on hold]

I'm trying to solve the following question : $$ maximize \ x^{2}-5\cdot x + y^{2}-3\cdot y $$ $$ x+y\leq 8 $$ $$ x\leq2 $$ $$ x,y\geqslant 0 $$ By using Frank Wolf algorithm ( according to ...
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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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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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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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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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(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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31 views

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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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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“constraint and column generation method” in Julia-Jump [closed]

Anyone has any idea of how to program "constraint and column generation method" in Julia-Jump for a mixed-integer three level problem optimization. thank you very much
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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. ...