Optimisation in finance .pdf

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Optimization is a branch of applied mathematics that derives its importance
both from the wide variety of its applications and from the availability of
advanced algorithms for the efficient and robust solution of many of its problem
classes. Mathematically, it refers to the minimization (or maximization)
of a given objective function of several decision variables that have to satisfy
some functional constraints. A typical optimization model addresses the allocation
of scarce resources among a set of alternative activities in order to
maximize an objective function–a measure of the modeler’s satisfaction with
the solution, for example, the total profit.
Decision variables, the objective function, and constraints are three essential
elements of any optimization problem. Some problems may lack constraints
so that any set of decision variables (of appropriate dimension) are
acceptable as alternative solutions. Such problems are called unconstrained
optimization problems, while others are often referred to as constrained optimization
problems. There are problem instances with no objective functions–
the so-called feasibility problems, and others with multiple objective functions.
Such problems are often addressed by reduction to a single or a sequence
of single-objective optimization problems.
If the decision variables in an optimization problem are restricted to integers,
or to a discrete set of possibilities, we have an integer or discrete
optimization problem. If there are no such restrictions on the variables, the
problem is a continuous optimization problem. Of course, some problems
may have a mixture of discrete and continuous variables. Our focus in these
lectures will be on continuous optimization problems. We continue with a
short classification of the problem classes we will encounter during our lectures.
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