I am new to heuristic methods of optimization and learning about different optimization algorithms available in this space like Gentic Algorithm, PSO, DE, CMA ES etc.. The general flow of any of these algorithms seem to be initialise a population, select, crossover and mutation for update , evaluate and the cycle continues. The initial step of population creation in genetic algorithm seems to be that each member of the population is encoded by a chromosome, which is a bitstring of 0s and 1s and then all the other operations are performed. GE has simple update methods to popualation like mutation and crossover, but update methods are different in other algorithms.
My query here is do all the other heuristic algorithms also initialize the population as bitstrings of 0 and 1s or do they use the general natural numbers?
The representation of individuals in evolutionary algorithms (EA) depends on the representation of a candidate solution. If you are solving a combinatorial problem i.e. knapsack problem, the final solution is comprised of (0,1) string, so it makes sense to have a binary representation for the EA. However, if you are solving a continuous black-box optimisation problem, then it makes sense to have a representation with continuous decision variables.
In the old days, GA and other algorithms only used binary representation even for solving continuous problems. But nowadays, all the algorithms you mentioned have their own binary and continuous (and etc.) variants. For example, PSO is known as a continuous problem solver, but to update the individuals (particles), there are mapping strategies such as s-shape transform or v-shape transform to update the binary individuals for the next iteration.
My two cents: the choice of the algorithm relies on the type of the problem, and I personally won't recommend using a binary PSO at first try to solve a problem. Maybe there are benefits hidden there but need investigation.
Please feel free to extend your question.
I am trying to parallelize gaussian elimination on sparse linear equations. I could not find data to test any where on the internet. If you could provide links to such data set that will be great.
Also could someone please explain how are sparse linear equations produced, that is practically, what problems produce such equations.
Thanks in advance.
First:
If indeed interested in linear problem sparse matrices, you may enjoy datasets from almost any linear Finite Element Method problem formulation.
Their problem-formulation may still use a vast amount of just linear equations.
Their problem-space is principally sparse in 3D-sense, so even if the original data were not represented as sparse-matrices, you can easily convert any of their dense-matrix representation into any sparse-matrix representation of your choice.
Next:
There are further sources of further interest for dense & sparse array I/O using the Matrix Market format
Background
I'm doing research about stability analysis of some dynamical system. In the literature, others have used analytical methods to query the signs of the eigenvalues of the linearized system matrix. My approach is to use a numerical eigenvalue solver. In some cases, input to the stability analysis consists of some coefficients that become infinite. In the analytical approach, this is tackled by taking the limit of the resulting stability criteria to infinity. "Taking the limit" is however not possible in a numerical approach, so I have reformulated the problem to avoid infinite coefficients in my numerical implementation.
Question
Now my question should be clear. Would a linear algebra package allow to use infinite coefficients? My direct application only needs eigenvalue solvers, but I don't want to narrow down to that. Any answer regarding infinite coefficients as input to linear algebra algorithms (matrix-solve, eigenvalue problem, singular value decomposition, LU etc.) is welcome.
I have solved a single objective convex optimization problem (actually related to reducing interference reduction) using cvx package with MATLAB. Now I want to extend the problem to multi objective one. What are the pros-cons of solving it using genetic algorithm in comparison to cvx package? I haven't read anything about genetic algorithms and it came about by searching net for multiobjective optimization.
The optimization algorithms based on derivatives (or gradients) including convex optimization algorithm essentially try to find a local minimum. The pros and cons are as follows.
Pros:
1. It can be extremely fast since it only tries to follow the path given by derivative.
2. Sometimes, it achieves the global minimum (e.g., the problem is convex).
Cons:
1. When the problem is highly nonlinear and non-convex, the solution depends on initial point, hence there is high probability that the solution achieved is far from the global optimum.
2. It's not quite for multi-objective optimization problem.
Because of the disadvantages described above, for multi-objective optimization, we generally use evolutionary algorithm. Genetic algorithms belong to evolutionary algorithm.
Evolutionary algorithms developed for multi-objective optimization problems are fundamentally different from the gradient-based algorithms. They are population-based, i.e., maintain multiple solutions (hundreds or thousands of them) where as the latter ones maintain only one solution.
NSGA-II is an example: https://ieeexplore.ieee.org/document/996017, https://mae.ufl.edu/haftka/stropt/Lectures/multi_objective_GA.pdf, https://web.njit.edu/~horacio/Math451H/download/Seshadri_NSGA-II.pdf
The purpose of the multi-objective optimization is find the Pareto surface (or optimal trade-off surface). Since the surface consists of multiple points, population-based evolutionary algorithms suit well.
(You can solve a series of single objective optimization problems using gradient-based algorithms, but unless the feasible set is convex, it cannot find them accurately.)
Several of my peers have mentioned that "linear algebra" is very important when studying algorithms. I've studied a variety of algorithms and taken a few linear algebra courses and I don't see the connection. So how is linear algebra used in algorithms?
For example what interesting things can one with a connectivity matrix for a graph?
Three concrete examples:
Linear algebra is the fundament of modern 3d graphics. This is essentially the same thing that you've learned in school. The data is kept in a 3d space that is projected in a 2d surface, which is what you see on your screen.
Most search engines are based on linear algebra. The idea is to represent each document as a vector in a hyper space and see how the vector relates to each other in this space. This is used by the lucene project, amongst others. See VSM.
Some modern compression algorithms such as the one used by the ogg vorbis format is based on linear algebra, or more specifically a method called Vector Quantization.
Basically it comes down to the fact that linear algebra is a very powerful method when dealing with multiple variables, and there's enormous benefits for using this as a theoretical foundation when designing algorithms. In many cases this foundation isn't as appearent as you might think, but that doesn't mean that it isn't there. It's quite possible that you've already implemented algorithms which would have been incredibly hard to derive without linalg.
A cryptographer would probably tell you that a grasp of number theory is very important when studying algorithms. And he'd be right--for his particular field. Statistics has its uses too--skip lists, hash tables, etc. The usefulness of graph theory is even more obvious.
There's no inherent link between linear algebra and algorithms; there's an inherent link between mathematics and algorithms.
Linear algebra is a field with many applications, and the algorithms that draw on it therefore have many applications as well. You've not wasted your time studying it.
Ha, I can't resist putting this here (even though the other answers are good):
The $25 billion dollar eigenvector.
I'm not going to lie... I never even read the whole thing... maybe I will now :-).
I don't know if I'd phrase it as 'linear algebra is very important when studying algorithms". I'd almost put it the other way around. Many, many, many, real world problems end up requiring you to solve a set of linear equations. If you end up having to tackle one of those problems you are going to need to know about some of the many algorithms for dealing with linear equations. Many of those algorithms were developed when computers was a job title, not a machine. Consider gaussian elimination and the various matrix decomposition algorithms for example. There is a lot of very sophisticated theory on how to solve those problems for very large matrices for example.
Most common methods in machine learning end up having an optimization step which requires solving a set of simultaneous equations. If you don't know linear algebra you'll be completely lost.
Many signal processing algorithms are based on matrix operations, e.g. Fourier transform, Laplace transform, ...
Optimization problems can often be reduced to solving linear equation systems.
Linear algebra is also important in many algorithms in computer algebra, as you might have guessed. For example, if you can reduce a problem to saying that a polynomial is zero, where the coefficients of the polynomial are linear in the variables x1, …, xn, then you can solve for what values of x1, …, xn make the polynomial equal to 0 by equating the coefficient of each x^n term to 0 and solving the linear system. This is called the method of undetermined coefficients, and is used for example in computing partial fraction decompositions or in integrating rational functions.
For the graph theory, the coolest thing about an adjacency matrix is that if you take the nth power of an adjacency Matrix for an unweighted graph (each entry is either 0 or 1), M^n, then each entry i,j will be the number of paths from vertex i to vertex j of length n. And if that isn't just cool, then I don't know what is.
All of the answers here are good examples of linear algebra in algorithms.
As a meta answer, I will add that you might be using linear algebra in your algorithms without knowing it. Compilers that optimize with SSE(2) typically vectorize your code by having many data values manipulated in parallel. This is essentially elemental LA.
It depends what type of "algorithms".
Some examples:
Machine-Learning/Statistics algorithms: Linear Regressions (least-squares, ridge, lasso).
Lossy compression of signals and other processing (face recognition, etc). See Eigenfaces
For example what interesting things can one with a connectivity matrix for a graph?
A lot of algebraic properties of the matrix are invariant under permutations of vertices (for example abs(determinant)), so if two graphs are isomorphic, their values will be equal.
This is a source for good heuristics for determining whether two graphs
are not isomorphic, since of course equality does not guarantee existance of isomorphism.
Check algebraic graph theory for a lot of other interesting techniques.