### Doctoral Thesis Defense, Optimization and Systems Theory

Monday, February 17, 2003, 10.00, Kollegiesalen, Administration building,
Valhallavägen 79, KTH

**Göran Sporre**

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On some properties of interior methods for optimization

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Akademisk avhandling som med tillstånd av Kungliga Tekniska
Högskolan framlägges till offentlig granskning för
avläggande av teknologie doktorsexamen måndagen den
17:e februari 2003 kl 10.00 i Kollegiesalen,
Administrationsbyggnaden, Kungliga Tekniska Högskolan,
Valhallavägen 79.
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This thesis consists of four independent papers concerning different
aspects of interior methods for optimization. Three of the papers
focus on theoretical aspects while the fourth one concerns some
computational experiments.

The systems of equations solved within an interior method applied to a
convex quadratic program can be viewed as weighted linear
least-squares problems. In the first paper, it is shown that the
sequence of solutions to such problems is uniformly bounded. Further,
boundedness of the solution to weighted linear least-squares problems
for more general classes of weight matrices than the one in the
convex quadratic programming application are obtained as a byproduct.

In many linesearch interior methods for nonconvex nonlinear
programming, the iterates can ``falsely'' converge to the boundary of
the region defined by the inequality constraints in such a way that
the search directions do not converge to zero, but the step lengths
do. In the second paper, it is shown that the multiplier search
directions then diverge. Furthermore, the direction of divergence is
characterized in terms of the gradients of the equality constraints
along with the asymptotically active inequality constraints.

The third paper gives a modification of the analytic center problem
for the set of optimal solutions in linear semidefinite programming.
Unlike the normal analytic center problem, the solution of the
modified problem is the limit point of the central path, without any
strict complementarity assumption. For the strict complementarity
case, the modified problem is shown to coincide with the normal
analytic center problem, which is known to give a correct
characterization of the limit point of the central path in that case.

The final paper describes of some computational experiments concerning
possibilities of reusing previous information when solving system of
equations arising in interior methods for linear programming.

Calendar of seminars

*Last update: January 10, 2003 by
Anders Forsgren,
anders.forsgren@math.kth.se.
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