Quizes Flashcards
A mathematical function f(x) has always only one global minimum. (TRUE or FALSE)
FALSE. Since a function can have any number of global minima or maxima. Consider, for example cos(x): This function has global minima (f = -1) at x = pi, 3pi, 5pi etc.)
Local optima occur only where f’(x) = 0.
FALSE. It is true that local optima occur where f’(x) = 0, but they can also occur at points where f’(x) does not exist.
What is the feasible set (In connection with constrained optimization)? (pick one answer)
A. The set of all points for which the function reaches its optimum.
B. The set of all points that fulfil the constraints.
C. The set of all points for which f(x) = 0
B. The feasible set is the set of all points that fulfil the constraints.
Which property applies to the Hessian matrix H (of a function f)? (pick one answer)
A. It is symmetric
B. Only the diagonal elements are non-zero
C. All its elements are always non-zero
A. The Hessian matrix is symmetric.
Consider a 2x2 matrix in which all elements are equal to 1. Is this matrix positive definite?
No. The equation for the eigenvalues will be: (1-lmbda)^2 – 1 = 0, with solutions: lambda_(1,2)= 1 ± 1, i.e. 0 and 2. Thus, not all eigenvalues are positive, meaning that the matrix is not positive definite. It is, however, positive semi-definite, since all eigenvalues are non-negative.
In an iterative method, once the search direction has been found, the problem of finding a suitable step length is one-dimensional.
Yes, this is TRUE. No matter the dimensionality of the vector x, once the search direction has been inserted, the resulting equation depends only on the step length (eta)
Newton-Raphson’s method will always converge to a local minimum. (TRUE or FALSE)
FALSE. Depending on the starting point, the method might converge to a minimum or a
maximum. Thus, once an optimum has been reached, one must check whether it is a minimum or a maximum (for example by considering the second derivative of f(x)).
An optimization problem is always convex if f(x) is convex. (TRUE or FALSE)
FLASE. It is necessary, but not sufficient, that f(x) should be convex. In order for an optimization problem to be convex, the constraints defining the feasible set S must fulfil certain criteria. More specifically, the inequality constraints must be convex, and the equality constraints must be affine.
Consider the level curves of a function f = f(x1,x2) and a constraint h(x1,x2) = 0. At the local optima of f (subject to h), the following holds: (pick one answer)
A. The gradients of f and h are perpendicular
B. The gradient of h is equal to the zero vector
C. The gradients of f and h are parallel
C. The gradients of f and h are parallel at the local optima.
Stochastic optimization methods can be applied even if the objective function f(x) is non-differentiable. (TRUE or FALSE)
TRUE, and this is one, among several, advantages with such methods. Stochastic optimization methods do not explicitly make use of gradients (or higher-order derivatives), and they can therefore handle non- differentiable objective functions.
If two individuals are able to have offspring, they must be of the same species. (TRUE or FALSE)
FALSE. Of course, the most common case is that the offspring is a result of the union of two individuals of the same species. However, individuals of some closely related (but different) species (e.g. lions and tigers) can have offspring. In that case, however, the offspring is not fertile, i.e. it represents an evolutionary dead end.
Acquired traits (i.e. things learned) can be transmitted to the next generation. (TRUE or FALSE)
No, this is (generally) FALSE. The transferable information is stored in the germ cells (sex cells) of an individual, and does not change during the lifetime of the individual. However, it is possible that changes that do not involve alterations in the sequence of base pairs, for example changes in gene activity (i.e. gene expression) can be transmitted (a concept known as epigenetics).
Humans have more chromosomes than any other known species. (TRUE or FALSE)
FALSE. Many other species (e.g. some species of fish) have more chromosomes.
Why must codons (sequences of bases coding for an amino acid) consist of
at least 3 letters?
This is so, since there are 20 (standard) amino acids. If the codon contained only two letters, they could encode at most 42=16 amino acids. With three letters, there are 64 possibilities, leading to some
redundancy.
All processes in Darwinian evolution are random and without direction. (TRUE or FALSE)
FALSE. Mutations are random (undirected), where selection most certainly is not. Selection is of course not predetermined (there is a stochastic element), but the process strongly favors fit individuals.
In an evolutionary algorithm, the fact that chromosomes are initialized randomly
slows down the process of finding the optimum. (TRUE or FALSE)
FALSE. In general, EAs are very fast in the early generations, quickly making up for their random starting point, normally in much shorter time than it would take for a person to assign some suitable non-random chromosomes.
It is important to make the right choice between binary and real-number encoding,
for a given optimization problem. (TRUE or FALSE)
FALSE. There is no evidence that one encoding scheme provides (generally) faster convergence than the other. Often, real-number encoding is used for simplicity, at least if the number of variables is large enough (more than 10, say) to allow crossover to operate as it should.
In tournament selection, the number of individuals participating in a tournament
is always equal to two. (TRUE or FALSE)
FALSE. The size of the tournaments can be generalized to any positive integer > 2 (up to the population size), even though two is the most common tournament size. See p. 50 for a description of how to implement tournament selection in cases where tournament size is larger than two.
