Mathematical programming

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In mathematics, computer science and economics, optimization, or mathematical programming, refers to choosing the best element from some set of available alternatives.

In the simplest case, this means solving problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. This formulation, using a scalar, real-valued objective function, is probably the simplest example; the generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, it means finding «best available» values of some objective function given a defined domain, including a variety of different types of objective functions and different types of domains.

Optimization problems are often multi-modal, that is they possess multiple good solutions. They could all be globally good (same cost function value) or there could be a mix of globally good and locally good solutions. Obtaining all (or at least some of) the multiple solutions is the goal of a multi-modal optimizer.

Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used to obtain multiple solutions, since it is not guaranteed that different solutions will be obtained even with different starting points in multiple runs of the algorithm. Evolutionary Algorithms are however a very popular approach to obtain multiple solutions in a multi-modal optimization task.

Adding more than one objective to an optimization problem adds complexity. For example, if you wanted to optimize a structural design, you would want a design that is both light and rigid. Because these two objectives conflict, a trade-off exists. There will be one lightest design, one stiffest design, and an infinite number of designs that are some compromise of weight and stiffness. This set of trade-off designs is known as a Pareto set. The curve created plotting weight against stiffness of the best designs is known as the Pareto frontier.

A design is judged to be Pareto optimal if it is not dominated by other designs: a Pareto optimal design must be better than another design in at least one aspect. If it is worse than another design in all respects, then it is dominated and is not Pareto optimal.

Part b. Science itself Unit 4. Did Darwin's Finches Do Math?

Task 1. Match the words with their definitions:

a beak	1a small amount or piece that is taken from something, so that it can be tested or examined; 2a single example of something, often an animal or plant
a finch	postdoctoral research fellow
to fine-tune	the hard pointed mouth of a bird [= bill]
a postdoc	a small bird with a short beak
a specimen	to make very small changes to something such as a machine, system, or plan, so that it works as well as possible
a curve	to try to make something fit into a space that is too small, or to try to get into such a space[= squash]; to press something firmly together with your fingers or hand
linear	1 the imaginary line around which a large round object, such as the Earth, turns; 2 a line drawn across the middle of a regular shape that divides it into two equal parts;3 either of the two lines of a graph, by which the positions of points are measured
to impose	1 if someone in authority imposes a rule, punishment, tax etc, they force people to accept it; 2 to force someone to have the same ideas, beliefs etc as you
a constraint	1 a line that gradually bends like part of a circle; 2 a line on a graph that gradually bends and represents a change in the amount or level of something
to scale	technical to make writing or a picture the right size for a particular purpose
to squeeze	something that limits your freedom to do what you want [= restriction]
an axis (PL axes)	a regularly repeated arrangement of shapes, colours, or lines on a surface, usually as decoration:
a gene	a plan, idea, or method that is feasible is possible and is likely to work:
feasible	1 consisting of lines, or in the form of a straight line
to conserve	1 to protect something and prevent it from changing or being damaged [= preserve]
a pattern	a part of a cell in a living thing that controls what it looks like, how it grows, and how it develops. People get their genes from their parents

Task 2. Look at the words and phrases given and group them into the table:

Fine-tuned, stout beak, natural selection, postdoc, specimen, curve, linearity, scaling, shearing, axes, gene expression, pattern formation, feasible, property

general science	biological science	mathematical science