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.
