Chapter 3 Quadratic Programming

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Optimization I; <strong>Chapter</strong> 3 73 with α chosen such that (x, µ, z) stays feasible. Since D µ is a nonsingular diagonal matrix, the increment ∆z in the slack variable can be easily eliminated resulting in ( B A T A −D −1 µ Z where g := −ZD µ e + σκe. ) ( ∆x ∆µ ) = ( −r b −r d − D −1 µ g ) , (3.69)
Optimization I; <strong>Chapter</strong> 3 74 3.7 Logarithmic barrier functions We consider the inequality constrained quadratic programming problem (3.62a)- (3.62b). Algorithms based on barrier functions are iterative methods where the iterates are forced to stay within the interior F int := { x ∈ lR n | a T i x − c i < 0 , 1 ≤ i ≤ p } . (3.70) Barrier functions have the following properties: • They are infinite outside F int . • They are smooth within F int . • They approach ∞ as x approaches the boundary of F int . Definition 3.7 Logarithmic barrier function For the quadratic programming problem (3.62a)-(3.62b) the objective functionals B (β) (x) := Q(x) − β p∑ log(d i − a T i x) , β > 0 (3.71) i=1 are called logarithmic barrier functions. The parameter β is referred to as the barrier parameter. Theorem 3.8 Properties of the logarithmic barrier function Assume that the set S of solutions of (3.62a)-(3.62b) is nonempty and bounded and that the interior F int of the feasible set is nonempty. Let {β k } lN be a decreasing sequence of barrier parameters with β k → 0 as k → ∞. Then there holds: (i) For any β > 0 the logarithmic barrier function B (β) (x) is convex in F Int and attains a minimizer x(β) on F int . Any local minimizer x(β) is also a global minimizer of B (β) (x). (ii) If {x(β k )} lN is a sequence of minimizers, then there exists lN ′ ⊂ lN such that x(β k ) → x ∗ ∈ S, k ∈ lN ′ . (iii) If Q ∗ is the optimal value of the objective functional Q in (3.62a)-(3.62b), then for any sequence {x(β k )} lN of minimizers there holds Q(x(β k )) → Q ∗ , B (β k) (x(β k )) → Q ∗ as k → ∞ . Proof. We refer to M.H. Wright; Interior methods for constrained optimization. Acta Numerica, 341-407, 1992. •
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Chapter 3 Quadratic Programming

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