Design optimization

Design optimization is an engineering design methodology using a mathematical formulation of a design problem to support selection of the optimal design among many alternatives. Design optimization involves the following stages:
Design optimization problem.
The formal mathematical (standard form) statement of the design optimization problem is 
formula_1
where
The problem formulation stated above is a convention called the "negative null form", since all constraint function are expressed as equalities and negative inequalities with zero on the right-hand side. This convention is used so that numerical algorithms developed to solve design optimization problems can assume a standard expression of the mathematical problem.
We can introduce the vector-valued functions
formula_11 " "
to rewrite the above statement in the compact expression
formula_12
We call formula_13 the "set" or "system of" ("functional") "constraints" and formula_9 the "set constraint".
Application.
Design optimization applies the methods of mathematical optimization to design problem formulations and it is sometimes used interchangeably with the term engineering optimization. When the objective function "f" is a vector rather than a scalar, the problem becomes a multi-objective optimization one. If the design optimization problem has more than one mathematical solutions the methods of global optimization are used to identified the global optimum.
Optimization Checklist 
A detailed and rigorous description of the stages and practical applications with examples can be found in the book Principles of Optimal Design.
Practical design optimization problems are typically solved numerically and many exist in academic and commercial forms. There are several domain-specific applications of design optimization posing their own specific challenges in formulating and solving the resulting problems; these include, shape optimization, wing-shape optimization, topology optimization, architectural design optimization, power optimization. Several books, articles and journal publications are listed below for reference.