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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Prove that is concave or convex

Please explain the solution step by step. Thanks :) !
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14 views

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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1answer
11 views

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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6 views

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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9 views

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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21 views

Maximum value of $x^TQx$ [on hold]

Can someone suggest me a clue for the following problem? What is the maximum value of $x^T Q x$ when $Q$ is a symmetric matrix and $x$ a vector of norm not exceeding 1? Thanks
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9 views

Generalization of minimisation problem

First I would like indtroduce my problem ! There is an easy way to solve this one : Find the value of $$ \inf_{(a,b)\in \mathbb{R}^2} \int_0^1 (t^2-at-b)^2 dt $$ and precise for which values $a$ ...
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12 views

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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2answers
2k 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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6 views

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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65 views
+100

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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206 views

Twilight Zelda Guardian Puzzle : Shortest Path (UPDATE: ADDED RULES)

I'm playing a video game right now and in it is a puzzle (see here). There are solutions to solving it (see here) on the Internet, but I'd like to know if this path is the shortest path (least amount ...
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9 views

matlab code for inexact PRP method for symmetric nonlinear equations [on hold]

pls how will write bactracking line search code in the paper inexact PRP method for symmetric nonlinear equations. pls help me with the codes
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16 views

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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19 views

Multiple constraints optimization problem in matlab

How can I solve an optimization problem with multiple constraints in matlab? I am trying to solve for: ...
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18 views

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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1answer
27 views

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 ...
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1answer
33 views

How can I optimize a multi-variable expression with a constant target.

I would like to know what methods are applied for optimizing multi-variable expressions with a defined target. I have a specific example I need help with, but I would like to be pointed into the ...
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12 views

Maximize a concave function under nonconvex constraints

I have to maximize the rate, which is a concave function, under certain constraints, where one of them is not convex; My optimization problem is: $\max_{\mathbf{P}_{2,n}} \frac{B}{L} \sum_{k=1}^L ...
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1answer
42 views

Which optimization class does the following problem falls into (LP, MIP, CP..) and which solver to use

I have the following optimization problem. I want to solve it using a computer solver. But I am not sure which problem class it falls into or which solver to use. Problem: There is a set of objects ...
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54 views

Finding $ \max_{x \in [2,4]} \left| 2 x \cos(2 x) - (x - 2)^{2} \right| $.

This is a problem taken from Burden’s and Faires’ Numerical Analysis. Define $ f: \Bbb{R} \to \Bbb{R} $ by $$ \forall x \in \Bbb{R}: \quad f(x) \stackrel{\text{df}}{=} 2 x \cos(2 x) - (x - 2)^{2}. $$ ...
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2answers
76 views

Maximisation: Half circle area inscribed within isoceles triangle?

Given an isosceles triangle with the line of symmetry along $x=0$, and the odd side along $y=0$, How can I optimize the maximum parabolic area inscribed within the triangle? The "half -circular" area ...
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1answer
908 views

How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
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1answer
15 views

How do I know that method of steepest descent works?

Here is the definition of the method of steepest descent given in the book "The mathematics of nonlinear programming" by Peressini. Suppose $f(x)$ is a function with continuous partial derivatives on ...
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1answer
52 views

finding extermal on old exam questions? [on hold]

I ran into a question that wants to find Extermal of following function: $$\int_0^2 \frac{ \dot{x}^2}{x^3} dt \quad \text{ with }\quad x(0)=1,\;x(2)=4$$ who can help me how we can solve this old ...
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13 views

Maximize the intersection over union of oriented rectangles

I have an oriented rectangle in the form region=(x1,y1, ..., x4, y4) I want to know which is the axis-aligned rectangle with the same center that maximize the intersection over union of the areas of ...
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13 views

l1 Quadratic Programming

Within a SQP- algorithm it can happen that the constraints of the quadratic sub- problems are infeasible. In order to overcome this infeasibilities, a l1 penalty method can be used according to ...
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17 views

Formulate a solvable optimization problem

I am trying to solve an optimization problem which could be temporarily formulated as follows, Objective: $\min \quad c_0(1-x_1)x_2x_3(1-x_4) + c_1x_1x_2(1-x_3)x_4 + c_2x_1(1-x_2)x_3(1-x_4)$ ...
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11 views

gradient descent to solve binary non linear optimization problem [on hold]

I am trying to code a solution for an optimization problem that has binary matrix which has to be optimized,since the problem is not convex and has binary variables,i am finding it hard to solve ...
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5 views

Minimizing wasted assignment of attributes to a person by optimising profiles and assigning them to each person

My maths is poor in this area but I'll try to be specific. I can solve this with brute force over the possible solution space but I'm wondering if I am ignorant of an algorithm, theory or approach ...
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1answer
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Solution of the LP relaxation - always round to the nearest integer?

If an optimal solution to the LP relaxation of an IP is not integer, can we always get a feasible IP solution by rounding it to the nearest integer? Or can we generalize this process by saying, if we ...
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199 views

mathematics of chemical stoichiometry

I would like to better understand the mathematical description of chemical stoichiometry and thermodynamic chemical equilibrium. This problem has many features and I know my description might be too ...
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0answers
14 views

Maximally distant orthogonal matrices

I would like to construct a set of $k$ orthogonal matrices in $\mathbb{R}^{n \times n}$ with maximal summed pairwise distance (in terms of L2 operator norm). Any ideas? I am thinking of just doing ...
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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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Heuristics for streaming data matching [migrated]

I have an index composed by thousands of documents. Slightly modified copies of those documents are sent to my application in small chunks, and I need to check, from those chunks, which document has ...
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13 views

Pseudoinverse with positive solutions

I'm not a mathematician but the engineering problem I'm considering is more of a mathematical question, that's why I post it here: Consider the matrices $M$ ($n \times 1$), $T$ ($n \times m$) and $F ...
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1answer
43 views

Minimum Volume of a circular, right cone, with a sphere inscribed in it.

Question: A sphere of radius $r$ is inscribed in a circular, right cone. What is the minimum radius and height of the circular cone? (Thus, volume) Because the answer would specifically ...
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26 views

Implementation of Lagrange Multiplier to solve constrained optimization problem.

I'm trying to solve an optimization problem. I have a list of around 4000 geo coordinates data, and I want to cluster them into 30 groups based on the distance, so that the closer properties belongs ...
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36 views

Finding maxima of a 3-variable function.

Let $x,y,z$ be positive real number satisfy $x+y+z=3$ Find the maximum value of $P=\frac{2}{3+xy+yz+zx}+(\frac{xyz}{(x+1)(y+1)(z+1)})^\frac{1}{3}$
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3answers
49 views

Finding the absolute maximum of the following 3d function

$ f(x,y) = \frac{(\lambda_1x+\lambda_2y+\lambda_3)^2}{x^2+y^2+1} $ I know that the function looks like some deformed dorito chip depending on the lambda values. That is about as far as I've gotten. ...
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1answer
441 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
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0answers
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What is an inner-outer iteration?

Inner-outer iterations are used in papers, for finding a stationary point of a system or in optimization. It is not clear, what is called an inner-outer loop though? Is it a nested loop where the ...
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0answers
12 views

Maximin optimization problem

I need to solve the following problem : Max.[ Min F(x,y ) ] where maximization is with respect to linear x , and minimization is with respect to non-linear y . The original problem had 6 ...
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Hierarchical Linear Programming

I am stuck with the following problem from research. For each time, $t$, I get a new data point $x_t$ and the current optimum value is a function of $\{x_t:t=1,2,\dots,T\}$ obtained by solving a LP. ...
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Minimising the surface area of a rectangular prism [Solution Verification]

A packaging company is going to make open topped boxes, with square bases that hold $100$ centimetres$^3$. What are the dimensions of the box that can be built with the least material?
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Find out the optimization type

I am formulating a problem and intend to solve it by optimization. Here is the current result: *Objective:*$\quad\min\quad c + f_1(x)x_1 + f_2(x)x_2$ Constraint: $\quad ax_1 + bx_2 <= d$ where ...
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1answer
24 views

Augmented Lagrangian Method for Inequality Constraints

Augmented Lagrangian Method can be used with inequality constraints. The question is how. One approach (according to Numerical Optimization Book by Nocedal and Wright; page 522), is linearly ...
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38 views

Find the maximum value of $x^{\alpha}y^{\beta}$ subject to the constraints $x+2y \le 2$ and $x > 0$ and $y > 0$.

The Statement of the Problem: Given real numbers $\alpha > 0$, $\beta > 0$, $\alpha + \beta \le 1$, find the maximum value of $x^{\alpha}y^{\beta}$ subject to the constraints $x+2y \le 2$ and ...
3
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1answer
139 views

Maximize the largest eigenvalue of a Hermitian matrix constrained by quadratic polynomials

I am looking for a method to maximize under $\mathbf{y}$ the largest eigenvalue of the following Hermitian matrix \begin{equation} S = \left [ \begin{array}{ccc} \mathbf{y}^{H}S_{11}\mathbf{y} ...
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17 views

Heuristic Optimization

I am working on a project with the NMF and additional cost terms. Therefor I am looking for an optimal weight factor for the cost terms to maximize the result. Because it is NP-hard and needs some ...