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

1. selection of a set of variables to describe the design alternatives;
2. election of an objective (decision criterion), expressed in terms of the design variables, which we seek to minimize or maximize;
3. determination of a set of constraints, expressed in terms of the design variables, which must be satisfied by any acceptable design;
4. determination of a set of values for the design variables, which minimize (or maximize) the objective, while being feasible, namely, satisfying all the constraints.

The formal mathematical statement of the design optimization problem is  

minimize f(x)

subject to h₁(x) = 0, h₂(x) = 0, ... , h_m1(x) = 0,

g₁(x) ≤ 0, g₂(x) ≤ 0, ... , g_m2(x) ≤ 0,

and x ∈ X ⊆ Rⁿ

where x is a vector of n real-valued design variables x₁, x₂, ..., x_n,  f(x) is the objective function, h_i(x) are m₁ equality constraints, g_i(x) are m₂ inequality constraints, and X is a set constraint that includes additional restrictions on x besides those implied by the equality and inequality constraints.  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 h = (h1, h2, . . . , hm1 )  and g = (g1,g2,...,gm2)T  to rewrite the above statement in the compact expression  

min f(x)

subject to h(x) = 0, g(x) ≤ 0, x ∈ X ⊆ Rn.  

We call h, g the set or system of (functional) constraints and X the set constraint.  

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.  Practical design optimization problems are typically solved numerically and many optimization software 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.