Question

Compare and contrast single and multivariate optimization. How are they alike? How are they different? Explain.

Answer 1

Optimization

By considering all the limitations and restrictions, making out the best alternative which can attain high performance with minimal cost is known as optimization. Optimization must be done by reducing the unwanted components and increasing the needed components. Hence, optimization does not consider the total expense incurred. It only aims at the maximum outcome.

Several kinds of optimization are there. Out of those, two types are discussed here. They are single optimization and multivariate optimization.

Single optimization (Single-variable optimization)

It is the optimization made with only one variable. It will be unconstrained, hence, also known as unconstrained optimization. In the case of single optimization, as there is only one variable, value can be taken as desired. There will be no constraints on determination of value. Hence, these optimization problems will take account of the issue of reducing an objective function which completely depends upon a single variable which does not have restricted value.

In case of certain applications, single-variable optimization will directly come to light. But, in some other cases, it will appear indirectly. Such indirect appearance will happen from reformulations of multivariate optimization.

Multivariate optimization (Multi-variable optimization)

It is the optimization made with several variables. It will be constrained, hence, also known as constrained optimization. In the case of multivariate optimization, as there will be more than one variable, value cannot be taken as desired. There will be certain constraints on determination of value. Hence, these optimization problems will take account of the issue of reducing or increasing an objective function which depends upon several variables which have restriction on its value. Multivariate optimization appears in those applications where clear restrictions are there on the variables.

Comparison of Single-variable optimization and Multivariate optimization

In case of certain applications, single-variable optimization will directly come to light. But, in some other cases, it will appear indirectly. Such indirect appearance will happen from reformulations of multivariate optimization. Whereas, multivariate optimization appears in those applications where clear restrictions are there on the variables. These constraints may differ in equality and inequality systems that depicts a complicated connections between the variables. Depending on the characteristic of the constraints, multivariate optimization problems can be divided into linear, convex and non-linear. Depending on the evenness of the constraints, multivariate optimization problems can be divided into differentiable and non-differentiable.