## ABSTRACT

Let x1, x2, …, xN be N decision variables in an optimization problem. In the general case, the problem is to search for values of the variables, x1, x2, …, xN, to maximize (or minimize) a given general function of the variables, subject to a given set of general constraints on the variables. The general formulation for the maximization problem is

( )1 2Maximize , , , Nf x x x… (13.1) subject to the M constraints

, , ,

, , ,

, , ,

g x x x b

g x x x b

g x x x b

⎫≤ ⎪⎪≤ ⎪⎬⎪⎪⎪≤ ⎭

…

…

…

(13.2)

where f(.) and gi(.) (i = 1, 2, …, M) are general functions bi (i = 1, 2, …, M) are constants

Expressing the decision variables in vector notation:

x

x

x

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

x

(13.3)

the general formulation for the maximization problem is written as

Maximize ( )f x (13.4)

subject to the constraints

( ) ( 1, 2, , )i ig b i M≤ =x … (13.5)

For the minimization problem, the corresponding general formulation is

Minimize ( )f x (13.6)

subject to the constraints

( ) ( 1, 2, , )i ig b i M≥ =x … (13.7)

A set of values for x that satis es all the constraints in a given problem is a feasible solution. The collection of all feasible solutions is the feasible search space.