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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Is this a discrete time Lyapunov function?

I have an algorithm to optimize a process. It is a discrete time algorithm. Every iteration of this algorithm changes the state of the process. I found a function, say $f(s)$, where $s$ is the state ...
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Minimization Problems & Collective Behavior

I am having some trouble with this problem: http://imgur.com/MX1aPzT Could anybody help me understand the concepts behind the question and how I should go about it? I have been reading through some ...
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maximize volume of a cone with a slant height [on hold]

Find the maximum volume of a cone with a slant height of 10 cm. $V=(πr^2h)/3$
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1answer
26 views

Minimum of sum of increasing and decreasing function

Suppose we have a function $f(x)$ defined for integer $x$ in some bounded interval, which is positive and increasing $$f(x+1)\geq f(x)\\ f(x)>0$$ , and a function g(x) which is positive and ...
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5 views

Datasource fetch interval optimization

I'm writing to you because I would like to find an starting point on what kind of algorithms to search or investigate to solve the following problem. So we have a limit on the amount of data we can ...
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1answer
33 views

Existence of global minimum

Could someone help me with this problem? Let $C$, $D$ convex and closed sets such that the intersection is empty. I want to show that the function $f: \mathbb{R^n} \to \mathbb{R}$ defined by $f(x) = ...
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3answers
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Maximize $x^2+y^2+z^2$ on $x^2+y^2+4z^2 = 1$

Hi this is a lagrangian optimization problem. Essentially as the title says, the question is asking us to maximize (if possible) $x^2+y^2+z^2$ on $x^2+y^2+4z^2=1$. I started by the standard ...
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Find the absolute maximum/minimum values of S(t) where S'(t) is a quartic function with lots of horrible decimal places.

So I have a problem where I'm to find the absolute maximum and minimum values of the following function... $S(t) = -0.00003237t^5 + 0.0009037t^4 - 0.008956t^3 + 0.03629t^2 -0.04458t + 0.4074$ ...
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1answer
34 views

Solving a minimization of the minimum problem

Let ${\bf c}_{1}$, ${\bf c}_{2}\in \mathbb{R}^{n}$, ${\bf A}\in\mathbb{R}^{m\times n}$ and ${\bf b}\in\mathbb{R}^{m}$. Show how one can solve the optimization problem: min ...
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1answer
23 views

Maximize $\sum_i \mathrm{rate}_i$ s.t. $\mathrm{rate}_i$

Question related to optimization problems. $$\mathrm{maximize} \sum\limits_{i=1}^{M}\log\left(1+f_i(\mathbf{x})\right)$$ ...
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22 views

find a minimum of a function

let $F(x)= \frac{2}{nx} + L^{2}e^{\frac{\alpha}{x^{r}}}$ for given $ n,L,\alpha$ show that $x^*=argminF(x) $ where $ x^*=\frac{\alpha^{\frac{1}{r}}}{(ln(n))^{\frac{1}{r}}}$ i know that F is convex ...
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1answer
29 views

Finding/approximating 2 unknowns using one equation

I’m doing experimental data in a chemistry lab and I have faced this mathematical problem at a point of my work. Hope you guys can help me with that. What would be the best way to find two constants m ...
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1answer
20 views

If $A + B = \frac{\pi}{3} (A,B>0),$ Then the minimum value of sec A + sec B is?

If $A + B = \frac{\pi}{3} (A,B>0),$ Then the minimum value of sec A + sec B is? I know the condition for minima but here there are two simultaneous variables , how and with respect to what do I ...
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1answer
15 views

Can Moore–Penrose pseudoinverse solve for underdetermined linear system?

Thanks for reading my thread. I am thinking, many of us know that Moore–Penrose pseudoinverse can solve for overdetermined system $Ax=b$, where $x=(A^TA)^{-1}A^Tb$; for exmplae the linear regression ...
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2answers
26 views

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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25 views

Inequality constraints in Lagrangian

I was reading Lagrangian multipliers . In the above text I can't understand why $\lambda \ge 0 $ for $g(x)\ge0$ and vice versa . Can anyone give me the explanation to this ?
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1answer
43 views

Got stuck with this $L^2(-1, 1)$ optimization problem. Any ideas where it comes from?

Statement Let $u(x) \in L^2(-1, 1)$. Solve the following optimization problem: $$ \begin{cases} J(u) = 4 \int_{-1}^{1} \sqrt{|x| (1 - |x|)} u(x) dx + \left(\int_{-1}^1 \sin(3\pi x) u(x) dx\right)^2 ...
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2answers
33 views

Maximize the distance between a point and a bounding rectangle

There are $n$ random points in the $x-y$ plane, whose coordinates are known beforehand. We can use a minimum bounding rectangle (MBR) to bound these points. In this scenario, the MBR can be rotated, ...
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23 views

Maximizing variance of Hamming distance of a system

I have a system as shown below, where 4 registers have 8 bit input A,B,C,...
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14 views

Interpreting constraints in an optimization problem

I am working on an optimization-based image denoising project in which I have three "flavors" of an optimization problem, one constrained and two unconstrained. They are given as follows: ...
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1answer
14 views

examination of the function

I need help.. Question An examination of the function $f:\mathbb{R}^2 \to \mathbb{R}$, $f(x,y) = (y-3 x^2)(y-x^2)$ will give an idea of the difficulty of finding conditions that guarantee that a ...
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2answers
42 views

maximizing a quadratic objective subject to sum constraint

Is it possible to solve the following problem $$ \max \,{x^ \top }x \\[0.1in] st: \sum\limits_{i = 1}^n {{x_i} = 1} \\[0.1in] 0 \le x_i \le 1 $$ Intuitively it happens when any one of the n x's is 1 ...
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27 views

How to find the minimal path between points in a planar set with holes in it?

When I was a commuter student, I would park in a very large parking that that had a set of stairs in a corner that I had to climb. In general, I had to park far away from this corner in an almost full ...
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Calculating $\arg\min_x (1-\Phi(x;\mu_1,\sigma_1^2)+\Phi(x;\mu_2,\sigma_2^2))$

I would like to find $x$ satisfying the following expression: $$\arg \min_x R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2)$$ where $$R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2) ...
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How to pick the leaving variable for Perturbation method? (Linear programming)

I am studying Optimization, a math course. We are going over simplex method and its variances. One of which is called the perturbation method. From this example, O is the objective function and ...
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1answer
20 views

Is a constrained optimization problem equalivant to its Lagrangian form?

For the following problem: $\text{min:}\ f(x)\\ s.t. \ g(x)\leq t$ Is the above problem equalivant to the following problem? $\text{min:}\ f(x) + \lambda g(x) \\ s.t. \ \lambda\geq0$ where $t$ and ...
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1answer
30 views

Givens rotation and retraction mapping

Here's part of the texts in the book Optimization Algorithms on Matrix Manifolds, when talking about the retraction on a matrix sub-manifold. A retraction is a mapping that maps a tangent vector in a ...
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2answers
56 views

Minimum value of $\sqrt{(1+1/y)(1+1/z)}$

If $y,z > 0$ and $y + z = c$ where $c$ is a constant, then what's the minimum value of $$\sqrt{\left(1+\frac1y\right)\left(1+\frac1z\right)}$$ I am having a hard time solving this.
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Determining maximum number of groups - maybe Linear Programming

Given a set D dogs, C cats, and B birds, for each dog d in D, there is a set c(d) which indicates the set of cats that dog d likes and a set b(d) birds that dog d likes. How do I find the maximum ...
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What's the solution for $\max_{x\in(0,1]}: \{-1-x\}$

What's the solution for the following optimization problem? Is the constraint set convex? $$\max_{x\in(0,1]}:\{-1-x\}$$
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Smooth approximation of maximum using softmax?

Look at the Wiki page for Softmax function (section "Smooth approximation of maximum"): https://en.wikipedia.org/wiki/Softmax_function It is saying that the following is a smooth approximation to the ...
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1answer
21 views

Are there known patterns among minimal expressions?

Let $R = F[z_1, z_2, \dots]$ be the finite-degree polynomials in a countable number of variables. Let $\mathcal{E}(R)$ be the set of all expressions of polynomials. Note that there could be an ...
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25 views

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? [closed]

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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3answers
40 views

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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42 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
67 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 ...