Optimization Problems - Problems Optimization problems in...

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Introduction–Optimization Problems Optimization problems in mathematics are problems in which we wish to minimize or maximize some real-valued function relative to some set of arguments. This latter set is often called the set of feasible alternaties or, simply the feasible set . The function to be minimized or maximized is variously called the cost function , the objective function or the criterion . In these lectures, we will deal almost exclusively with cost functions, f o , which are defined on subsets of R n . In this case, the feasible set is often defined, at least partially, by a set of inequalities or equalities. Such explicit constraints are given in terms of a number of real-valued functions f i : R n R , i = 1 , 2 , . . . , m . But it may well be the case that constraints are given more abstractly in terms of some prescribed subset K ⊂ R n . This type of description is particularly useful in treating constraints of a geometric type, or, for example, range constraints in which we prescribe feasible points as ones whose components lie within certain upper and lower bounds, or, quite commonly, to keep the components of the feasible set non-negative, x i 0. More explicitly, we can write such a problem as minimize f o ( x ) subject to x ∈ K f ( x ) b g ( x ) = c . Here, the vector x = ( x 1 , x 2 , · · · , x n ) is the optimization variable of the problem 1 . The function f o : R n R is the cost function, the components of the function f : R n R m , namely f i : R n R , are constraint functions describing inequality constraints, and the component functions, g i : R n R , of g : R n R k describe equality constraints. The set K ⊂ R n is a (geometric) constraint set. Definition 1.1 A vector x will be called optimal if it gives the smallest value to f o amongst all vectors x satisfying the constraints, i.e., amongst all feasible vectors. We need to make some simple observations regarding this standard form. First, we have stated the problem as a minimization problem. Since maximizing a function over some 1 Vectors will always be considered as column vectors. The symbol here designates the operation of transpose which interchangs columns and rows. 1
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feasible set is equivalent to minimizing the negative of that function, we do not lose any generality in considering only minimization problems. Furthermore, any inequality constraint of the form h ( x ) d can be rewritten as - h ( x ) ≤ - d
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