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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A clarification regarding dynamic programming.

This is a question regarding dynamic programming. The document to which I am referring is this (pg 325). It says that $$v_n(s_n)=\text{Min}\{t_n(s_n)+v_{n-1}(s_{n-1})\}$$ Here $v_n(s_n)$ is the ...
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Iterative method to compute only the positive eigenvalue's and corresponding eignevectors of a very large matrix?

I have a very large dense matrix (~10000 X ~10000) which is not full rank . I want to compute only the positive eigenvalues and corresponding eigenvectors instead of computing all of them. I have ...
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36 views

Simple question on maximizing a family of concave functions

Given $x \in [0,1]$, let $f_k(x)$ be concave function in $x$ for all $k= 0,1,...,N$. Is the following two maximization formulations equivalent? $$ \max_{0 \le k \le N} f_k(x) =?= \max \{f_0(x), ...
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40 views

Can I perform Maximum likelihood via optimization?

I have two $3 \times 3$ matrices $\mathbf{a}$ and $\mathbf{f}$. $\mathbf {f}$ is completely known to me. Also $a_{ij} \in [+1,-1]$ \begin{equation} \mathbf{f} = \left( \begin{array}{ccc} f_{11} ...
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Can a Convex QCQP Problem with an additional linear constraint be converted to a SOCP?

I have a quadratically constrained quadratic programming problem that I massaged into the form $$ \begin{aligned} & \underset{x}{\text{minimize}} & & x^T Q x \\ & \text{subject to} ...
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22 views

A length of fence encloses an area alongside a river — what is the optimal shape to maximize area? [duplicate]

You have $100$ meters of fence . There is a perfectly straight riverbank, much longer than $100$ meters, so you have plenty of room to work with. What is the optimal shape and dimensions that ...
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18 views

Optimization: maximizing nonconvex sum of product of constraints

I'm wondering if there is any way to convexify, approximate, and/or simplify the following problem. $\max. \sum_{k \in K} \prod_{i \in I} (a_{ik} x_{ik} + b_{ik})$ s.t. $x_{ik} \in [0,1]$ where ...
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How should I modify the initial guess for normalised version of an optimisation problem?

I am solving a large optimisation problem. The problem contains 500 decision variables and few hounderd constraints. Since the constraints are very tight and nonlinear, any random initial guess ...
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144 views

Minimum value of $\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\frac{24}{5\sqrt{5a+5b}}$

Let $a\ge b\ge c\ge 0$ such that $a+b+c=1$ Find the minimum value of $P=\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\dfrac{24}{5\sqrt{5a+5b}}$ I found that the minimum value of $P$ is ...
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Product expression maximization given 4 matrices as function of 2 column vectors

Hypothesis: - we are given 4 complex matrices denoted with $H_1, H_2, G_1$ and $G_2$. - the 4 matrices are not necessarily square so their size is $N$ by $M$. - we denote with $w_1$ and $w_2$ two ...
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A repository of constrained optimization test problems?

I am looking for a repository of constrained optimization problems with solutions. I want to find "benchmark" type problems to test my algorithm on and just trying to search for known problems ...
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31 views

What is topological learning?

I am getting this term topological learning in few places for example a reference is below at section 1.1.2: http://virenjain.org/thesis/VirenJainThesis_official.pdf Can anyone point out what ...
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68 views

How to solve the Few Scientists Problem (big word problem) in its general form?

I'm trying to figure out how to solve this word problem. I'm pretty sure it involves calculus or something even harder, but I don't know how to solve the general form. Let me start with the concrete ...
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35 views

Optimisation Problem for Pipe Nesting

I work in a company where we are supposed to produce and send pipes using trucks to buyers. Pipes of smaller diameter can be nested inside pipes of larger diameter while sending to minimize number of ...
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Convex Optimization Problem [on hold]

Can anyone please solve the following optimization problem? it should be solvable using lagrangian relaxation method.
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Given the Hessian of a scalar function of two variables, find the function

Suppose we are given the Hessian of a scalar function of two variables, $\nabla^2_{f(x,y)}$, do there exist explicit formulas for finding $f(x,y)$? Attempt: Brute-Force method is to do double ...
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Nocedal/Wright: Numerical Optimization, Lemma 12.3.(ii)

In the above given monograph (1999, 1E) the following parametrized system of equations $R:\mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^n$ is introduced: $$ R(z,t) := \left[ \begin{array}{c} ...
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Choosing a surface that makes the flux of F maximal,

For a closed surface S in $R^3$, consider the flux of F, given by the usual flux integral. For what choice of S will the flux be maximal? So, I want to apply the divergence theorem and instead look ...
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103 views

Ideal amount of piles to sort a stack of 250 cards (magic the gathering)

I'm a hobbyist working on a mechanical sorting machine to sort magic the gathering cards. I'm by no means a mathematician though, and I was wondering if you all wouldn't mind helping me out with a ...
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28 views

Quasi concavity and quasi convexity of a max function

Consider the production function: Q = max {K, L}. We have to find out whether this function is quasi concave or quasi convex. According to me, since this function is concave, it should automatically ...
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Solving a matrix equation using numerical optimization

To my knowledge, if $A \in \mathbf{S}^n_{++}$, then given any $b \in \mathbb{R}^n$, the system of linear equations $Ax = b$ has a unique solution $x^* \in \mathbb{R}^n$. Moreover, the solution $x^* ...
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9 views

Ensuring parameters of log-linear model sum 1

I am training a log-linear model with parameters $\theta$ using SGD. I want to ensure that my parameters will end up being probabilities i.e. $\sum_i \theta_i = 1$. One way to do this is by using ...
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Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability [duplicate]

Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability $f:\mathbb{R}^n\rightarrow\mathbb{R}$ I don't know how to star this problem, I just know the definition ...
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Taylor series approximation of function under norm

I am reading this paper. At page number $4$, term $||Au - f||^{2}$ is approximated by taylor series approximation around $u^{k}$. The resulting approximations are $$\|Au - f\|^{2} \approx ...
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Simple verification: is this equivalence always true?

I have a constrained optimization problem and I am trying to reduce the number of contraints and am afraid to be losing information by doing so. If we have two constraints as the following $$A \geq B ...
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How to find a minima for a function

I'm trying to find a minimum value for the function $[(4N)^{r-q-2} r(r-1)...(r-q-1)]^\frac{1}{2^{q+2}-2}$ where $N$ is an integer, $r$ may or may not be an integer and $q$ is an positive integer ...
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21 views

Gradient of a forth order scalar function with respect to a Matrix

I'm trying to take the gradient of the following function w.r.t A: $$ f(A) = ||AC_YA^T - C_R||_F^2 $$ I tried the following: $$ f(A) = trace((AC_YA^T - C_R)^T(AC_YA^T - C_R)) = ...
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If the iteration $x^{k+1}=x^k-t_kH_k^{-1}\nabla f(x^k)$ converges superlinearly to a stationary point $x^*\ne x^k$, then $t_k\to 1$

Let $f\in C^2(\mathbb{R}^n)$ $(H_k)_{k\in\mathbb{N}_0}\subseteq\text{GL}_n(\mathbb{R})$ $x^0\in\mathbb{R}^n$ and $$x^{k+1}:=x^k+t_k d^k\;\;\;\text{for }k\in\mathbb{N}_0\tag{1}$$ with ...
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Max flow min cut: value of a flow

Consider a network $N$ with lower bound $a_{ij}$ and upper bound $b_{ij}$ and assume that N has a feasible flow. Now I have to prove that the value of a max flow is equal to the capacity of a min ...
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In an ODE dynamic system, is there a convient way or algorithms for estimating the parameters which make the ODE solution satisfing some constraint?

I have construct a ODE dynamic system like this $$molA(t)==sa$$ $$molB'(t)=sb-db\;molB(t)+\frac{kab\;molA(t)\;molB(t)}{molB(t)+Jab}-\frac{kgb\;molG(t)\;molB(t)}{molB(t)+Jgb} $$ $ molC'(t)=sc-dc\ ...
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Setting up the equation for optimization problems

Micheal is on an island and needs to get to a bank on shore, He knows that he can swim at 3km/h and jog at 10km/h. The island is 1500m from the shore and the bank is 800m from the point on the shore ...
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Minimize $f(m)=\sum_{n=1}^\infty n^m / m^n $

For what real value of $m$ such that $\displaystyle\sum_{n=1}^\infty \frac{n^m}{m^n} $ is minimized? I've been told that it's equivalent of solving for $ \text{Li}_{-n}\left(\frac1n\right)$ for the ...
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Minimize cost function, provide algebraic expression and determine optimal input sequence if $N = 4$

Given the discrete-time linear system $x(t+1) = 2x(t) + u(t)$ with $x(0) = 0$ and $t\geq0$, the goal is to find an optimal input sequence $u^{*}(0), \dots, u^{*}(N-1)$ that minimizes the following ...
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A Maximization Problem involving Fourier Coefficient Constraint

Let $f$ be a probability density function defined over the interval $[-1/2,1/2]$. For notational simplicity, we represent the inner product $\langle f\mid g\rangle=\int_{-1/2}^{1/2} f(x) g(x) dx$. ...
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optimization function

If we want to maximize a function $f$ and we know that $f$ depend on 3 variables $v_1$, $v_2$ and $v_3$ without knowing the exactly relation between them but we know that maximizing $f$ can be ...
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68 views

Lagrange multipliers problem

I have a two variables function: $f(x,y)=3x+y$ and I wish to find its minimum and maximum values with the constraint $\sqrt{x} +\sqrt{y} =4$. According to the answer, there is a minimum and a maximum. ...
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weakly compactness and a minimization

Suppose $S\subset L_1 $ is weakly compact with respect to weak topology. $$g^*: S\to L_{\infty}$$ We have known for every $f\in S$, $g^*(f)\in A$. Suppose $$\inf_{f\in S} F(f,g^*(f))$$ is not ...
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Prettifying Bipartite Graph Matrix

Let me tell you the story of this problem. We have $n$ projects and $m$ workers. Each worker can work on multiple projects and each project can be solved by multiple workers. This relationship can be ...
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Polynomial roots finding algorithm

My initial problem is a parameter estimation problem that is solved by minimining a least-square criterion with the Gauss-Newton algorithm. However finding a good initial iterate is very tedious. ...
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Property of the local solution of a static constrained optimization problem

In Nocedal/Wright's Numerical Optimization (1999, 1E) on p. 332 in subsection Feasible Sequences of section 12.3 Derivation of the First-Order Conditions they claim that a local solution $x^*$ to a ...
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Asset optimization using Excel

I originally posted this in superuser, but was told it was more of a math problem than an Excel problem. I'm working on a project, part of which involves figuring out the number and type of vessels ...
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Fencing optimization question that seems to be underspecified

I just had a test question that asked the following (I wrote down the question verbatim): A rectangular lot is going to be built adjacent to a road. The fencing next to a road costs $15$ dollars ...
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What is the behavior of the spatial median in high-dimensional spaces?

I am a photographer who is investigating a technique known as image stacking, in which multiple images of the same subject are combined to reduce noise (by CLT). Commonly used techniques are mean and ...
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Maximal Triangle Partitioning in n lines

Recently I was given the following problem at work: Given a 5 pointed star, draw two straight lines through it so that there are 10 minimal triangles within the drawing. It took some work but I ...
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Optimization problem: smallest euclidean distance with positive entries constraints

Suppose there is the simple function: \begin{align} f(x,y,z) &= (x-a)^2 + (y-b)^2 + (z-c)^2 + (x+y-S-z - d)^2 \end{align} where $a,b,c,d$ are nonnegative constants, and $S$ is an integer. I ...
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optimize the volume of a box where the sum of the h*d*w is the only known variable

I'm currently stuck with the following optimization excercice: Consider a box such that sum of its sides is $210$. Find the maximum volume that the box can have. Here are my current thoughts: ...
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Finding min and max under constraints

I have a two variable function: $f(x,y)=4x^2-y^2-xy-2x+6y$. I need to find its absolute minimum and maximum under the constraints: $y=4-2x$, $x \geq 0$, and $y \geq-2$. I am not sure how to do it, ...
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51 views

Closed Form Solution for Minimization involving Standard Normal CDF and PDF

Could someone please advice and provide detailed steps regarding any possible closed form solutions or other suggestions regarding solving a minimization problem of the type shown below? Here, $\phi$ ...
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What is the minimum point of $x\mapsto x^Ty$ for $|x|\le 1$ and a fixed $y\in\mathbb{R}^n$?

Let $y\in\mathbb{R}^n$. I want to minimize $$f(x):=x^Ty\;\;\;\text{for }|x|\le 1$$ The minimum point should be $$-\frac{y}{\sqrt{y^Ty}}\tag{1}$$ However, how can we derive $(1)$ analytically? Since ...
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Making projected search directions conjugate

I'm trying to implement a minimization process for the optimization problem: ...