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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Convex Subset Projection

Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ...
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Formal definition of convexity for multivariate function?

Let $M\in R^{M\times N}$, a function $f: M\rightarrow R$ is called convex on $M$ if $f\big((1-\lambda)X1+\lambda X2, (1-\lambda)Y1+\lambda Y2\big) \leq (1-\lambda)f(X1,Y1) + \lambda f(X2,Y2)$ For ...
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How is the upper bound of a minimisation IP determined during branch-and-bound?

When using the branch-and-bound algorithm to solve an Integer Programming (IP) problem, the entire enumeration tree doesn't need to be evaluated and that's where the speed-up is achieved. Suppose the ...
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max and min values on symmetric polytope

Let $-N\leq t \leq N$. Let $A$ be regular $(N-1)$-dimensional simplex with vertices $(t,0, \ldots, 0)\ldots (0, 0,\ldots, t)$ and $B$ be regular $(N-1)$-dimensional simplex with vertices $(t-N+1,1, ...
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How to find fitting parameters of the function?

I have the function describing the experimental data - $f(x)$. I also have another function - $g(x, \bar{p})$, which is the theoretical function for the process involved. Here $\bar{p}$ - is the ...
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Optimizing single element of vector wrt. second order cone constraints

Can anybody put me on the right path to solving the following problem analytically: Given a vector $\bf{x}=(x_1,...,x_n)^T$, how do I find the bounds for a single element subject to second order cone ...
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Optimize profit given complete market information

Assume there are $N$ market participants (on the order of several hundred), and $M$ items (several thousand) being bought and sold on a market. For each participant/item pair, you know how many units ...
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No critical points means convex or conave? [on hold]

If we don't know whether $f(x)=0$ is convex or concave or not, but we know under certain constraint sets there is no critical points of $f(x)$ inside meaning the solution of $df(x)=0$ is outside the ...
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Optimizing over a set of optimization problems

This is my first time asking an optimization question on here, so I am looking forward to see what will happen here. In the lack of a better title, I wrote it as it is. At a high-level, I can perhaps ...
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making a function non-linear using a Lagrangian function

How Is this formula a Lagrangian function ? And how can a non-linear element be added to a function using this "Lagrangian function" This is where i got this In order to improve the performance ...
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how does probabilistic optimization differs with other types of optimization?

probabilistic optimization and other types of optimization are very confusing. how can you differentiate it? i would want to optimize an equation using MATLAB. however, i do not know which or where ...
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Issue with CVX geometric programming

So I'm trying the following geometric optimization problem in CVX and I'm running into this weird issue where I get a higher optimal value if I remove a constraint. Here's the code I have run. The ...
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Ladder Optimization Problem

A fence 4 feet tall runs parallel to a tall building at a distance of 4 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of ...
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37 views

Minimize Function over Convex Subset

Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ...
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3answers
58 views

How to compute Lipschitz Constant for multivariate function $f(x,y)=1-xy$?

How to compute Lipschitz Constant for multivariate function $f(x,y)=1-xy$? I know the definition for one variable? What is its definition for multivariate functions?
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Golden search method iterations and minimum.

Theoretically, how many iterations should it take to minimize f to be within 〖10〗^(-m) over [a,b] using the Golden Search Method?
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Lagrangian Method Proof

Suppose $f(\mathbf x)$, $g(\mathbf x)$ are smooth functions where $\mathbf x^*$ is a constrained local minimizer of $f(\mathbf x)$ subject to $g(\mathbf x)=0$. If $\nabla g(\mathbf x^*) \neq 0$ and ...
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arc wise connected set

I am having confusion in understanding what is arc wise connected set.The definition is a set $S$ is arc wise connected if for any pair of point a,b we can define a continuous function $f$ from ...
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maximize the acos function of more than one variable

I want to find the maximum angle which is defined as follows: $\theta = \cos^{-1}(\frac{1 + x_1 + x_2}{\sqrt{1 + x_1^2 + x_2^2}})$ now I should find $x_1$ and $x_2$ values so that $\theta$ has its ...
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Maximizing the trace

Say i have the following maximization. $ max_R$ trace $(RZ): R^TR = I_n$ where $R$ is an $n$ x $n$ orthogonal transformational vector. Also, the SVD of $Z = USV^T$. I'm trying to find the optimal ...
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Regularized least squares

In Image Restoration, a true image f (in vector form)can be related to degraded data y through a linear model of the form $$y = Hf + n$$ where H is 2d blurring matrix and n denotes noise vector and ...
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Find max and min of $F=ax^2+2bxy+cy^2$ when $x^2+y^2=1$

Find the Maximum and Minimum of $$F=ax^2+2bxy+cy^2$$ when $$x^2+y^2=1$$ The variables a,b, and c are just real numbers. I have attempted using partial differentiation in order to solve for the ...
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38 views

Convex Functions and Subsets

Suppose that $f, g: \mathbb R^n \to \mathbb R $ are $C^1$ convex functions. Show that $C = ${$\mathbf x \mid g(\mathbf x) \leq 0$} is a convex subset of $\mathbb R^n$. Show that if $\nabla f(\mathbf ...
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Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
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minimizing sum of squares

Say i have the following optimization problem. min $\sum\limits_{i=1}^m \parallel r - (y_i - Rx_i) \parallel_2^2$. where we are optimizing over $r \in R^n$ and also $R \in R^{n, n}$ is given. Also, ...
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Linear Programming, Optimal Solutions

I posted the whole question to give some context, but my problem lies with (iv). I think you're meant to use a formula for the generalization of the optimal solution, but I'm not really sure what ...
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Optimization issue, how to obtain the maximal value?

$ max f(\beta)=\frac{\beta}{1+\beta}\cdot \left(1- \frac{\binom{N+B}{B}\cdot\beta^B} {\sum_{i=0}^B {\binom{N+i}{i} \cdot \beta^i}} \right)$ where $\beta\in[0,\infty)$, $N$ and $B$ are identified ...
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Inequality optimization, KKT condition.

So we have the problem: maximize $x^2+y^2$ subject to $x^2-y \leq3$ and $y\leq 1$. And I sorted out the KKT conditions for the problem (is here where the problem is?): $2x=\lambda _12x$, ...
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36 views

Maximize arccos-function

I need to find a maximum of the function $$y=\arccos\left(\frac{29+12x\sin(22)+6x\cos(22)+x^2} {\sqrt{x^2+6x\cos(22)-20x\sin(22)+109}\sqrt{x^2+6x\cos(22)-4x\sin(22)+13)}} \right) $$ between x=0 and ...
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A function to maximize a variable and minimize the other partially.

So I need a function that has two input variables $x$ and $y$. Let' say that $x$ is a variable representing a score. And $y$ is a variable representing steps taken to reach that score. I need a ...
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Understanding optimization on non-compact region

Say we have $f(x,y) = x^2 e^{-x^2 - y^2}$ and we want to optimize it over $\mathbb{R}^2$. The minimum value is $0$ since $f(x,y) \geqslant 0$; the question is whether a maximum value exists or not. ...
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The minimum of $\dfrac{x(1+y)+y(1+z)+z(1+x)}{\sqrt{xyz}}?$

Can we use $AM\geq GM$ inequality to find the minimum of $\dfrac{x(1+y)+y(1+z)+z(1+x)}{\sqrt{xyz}}?$ I can find out that minimum is $6$, but can we use $AM\geq GM$ to show this?
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Google Code Jam's Cookie Clicker Program…

Today, the Google Code Jam's cookie clicker problem was something like this. Problem In this problem, you start with 0 cookies. You gain cookies at a rate of 2 cookies per second, by ...
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How do you find a minimum of a function with these tools?

Let's say I can define a group $G$ acting on a set of combinatorial objects $X$ and I have a function $f: X \to \Bbb{N}$ that I want to find a minimum of in $X$. Is there a polynomial time ...
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Linear programming to find minimal additive and multiplicative factors

Consider samples $\{x_i,y_i\}$ with $x_i\in\mathbb{R}^N$ and $y_i=\pm1$ and additional $z\in\mathbb{R}^N$. Can one use linear programming to find the minimal $m>0$ and minimal $\epsilon>0$ (e.g. ...
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Mean value theorem mindset

So I am to learn to use the mean value theorem to prove these types of problems that I will list. I would really like for someone to provide some visual/intuitive information on how I can imagine the ...
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Find extrema on the interval

Problem Find the extrema of the function $$f(x) = cos^2(x)$$ on the interval $ [-4,4]$ I can differentiate and get $$f'(x) = -2 \sin(x) \cos(x)$$ And set that to zero, but I'm pretty sure that's ...
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How to find the maximizer in the Legendre transform of cumulant generating function?

Cramer's theorem in the large deviations theory gives the rate function $\sup_{\boldsymbol{\lambda}} \left<\boldsymbol{\lambda},\,\mathbf{b}\right>-\log\int_{\mathbb{R}^n} ...
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Optimization problem (total distance from point on sphere to other points)?

Let $M_i(x_i, y_i, z_i)$ be a set of $n$ fixed points. Given their coordinates, find a point $M(x, y, z)$ which is on the sphere $x^2 + y^2 + z^2 = 1$ and has the minimal sum of distances between ...
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Find min of $\frac{\left( \sum kx_k \right) \left( \sum x^2_k \right)}{\left( \sum x_k \right)^3}$

With $n \ge 2$ and $x_1,\ x_2,\ \dots,\ x_n > 0$. Find the minimum of: $$ M = \frac{(x_1 + 2 x_2 + ...+ nx_n)( x^2_1 + x^2_2 +...+x^2_n)} {\left( x_1 + x_2 +...+ x_n \right)^3}$$ For specific ...
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Minimizing sum of squared distances from point to spheres

Given some spheres with known radius and known origin in three dimensional space, I want to find the point P that lies "closest" to all these spheres. The meassure of closeness, I guess, will be the ...
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Making sense of the big world of gradient methods

There are many extensions of gradient descent: stochastic-, Nesterov accelerated-, proximal-, conjugate-, dual-, mirrored-, splitted-, coordinate- gradient descend and more. It also appears that many ...
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Positivity of a function in $\mathbb{R}^{n}$

We place ourself in $\mathbb{R}^{n}$. We consider a given increasing function $$ g : \begin{aligned} &\mathbb{R}^{+} \to \mathbb{R} \\ &x \;\;\,\mapsto g(x) \end{aligned}$$ Finally, we ...
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Quickly checking if an inequality holds on a convex region

Let $C$ be a given convex polygon in $\mathbb{R}^2$ containing the origin and let $a$, $\mathbf{b}$, and $Q\succeq0$ be a given scalar, vector, and matrix respectively. Is there a fast way to verify ...
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Model If-Then-condition [closed]

Is there a possibility to model the condition: "If x > 0 , then y = 1" with linear (in-) equalities, if $x \geq 0$ and $ 0\leq y \leq 1 $ and integrality is not allowed in the model? If I could ...
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1answer
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Simple Lagrange Multiplier Problem, not working out

The question should be simple. Use the Lagrange Multiplier to maximize $f(x,y) = 4x^2 + 10y^2$ subject to the constraint $x^2 + y^2 = 4$. But when I set it up I get two different values for ...
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Dual of a Semi Definite Programming Problem

How do I write the dual of the following semi definite programming problem? \begin{align} \max_{\lambda,y_i}~&\lambda \\ &\sum_{i=1}^{L}y_i\mathbf{C}_i-\lambda\mathbf{I}\geq 0 \\ ...
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The minimum distance from the circle $x^2+(y+6)^2=1$ to parabola $y^2=8x$?

What are the coordinates of the points on the parabola $y^2=8x$ which are at the minimum distance from the circle $x^2 + (y+6)^2=1$?
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Determining a value $c$ such that the function $f(c) = \displaystyle\sum_{i=1}^n \left|\frac{y_i-c}{y_i}\right|\times v_i$ is minimized

I'm trying to construct prediction model for a variable of interest, based on a set of input values. I have a set of validation data and their predictions (by my model) and now I need to asses whether ...
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34 views

Evaluate smartly a function on a multiplication grid

I am asking myself the following question: Suppose one has a grid $G \in \mathbb{N}^{n\times n}$ where $g_{ij} = i\cdot j$, $i,j \leq n$. I would like to evaluate a function $f: G \to \mathbb{N}$. ...