5 research outputs found
Global convergence of a stabilized sequential quadratic semidefinite programming method for nonlinear semidefinite programs without constraint qualifications
In this paper, we propose a new sequential quadratic semidefinite programming
(SQSDP) method for solving nonlinear semidefinite programs (NSDPs), in which we
produce iteration points by solving a sequence of stabilized quadratic
semidefinite programming (QSDP) subproblems, which we derive from the minimax
problem associated with the NSDP. Differently from the existing SQSDP methods,
the proposed one allows us to solve those QSDP subproblems just approximately
so as to ensure global convergence. One more remarkable point of the proposed
method is that any constraint qualifications (CQs) are not required in the
global convergence analysis. Specifically, under some assumptions without CQs,
we prove the global convergence to a point satisfying any of the following: the
stationary conditions for the feasibility problem; the
approximate-Karush-Kuhn-Tucker (AKKT) conditions; the trace-AKKT conditions.
The latter two conditions are the new optimality conditions for the NSDP
presented by Andreani et al. (2018) in place of the Karush-Kuhn-Tucker
conditions. Finally, we conduct some numerical experiments to examine the
efficiency of the proposed method
Encoding inductive invariants as barrier certificates: synthesis via difference-of-convex programming
A barrier certificate often serves as an inductive invariant that isolates an
unsafe region from the reachable set of states, and hence is widely used in
proving safety of hybrid systems possibly over an infinite time horizon. We
present a novel condition on barrier certificates, termed the invariant
barrier-certificate condition, that witnesses unbounded-time safety of
differential dynamical systems. The proposed condition is the weakest possible
one to attain inductive invariance. We show that discharging the invariant
barrier-certificate condition -- thereby synthesizing invariant barrier
certificates -- can be encoded as solving an optimization problem subject to
bilinear matrix inequalities (BMIs). We further propose a synthesis algorithm
based on difference-of-convex programming, which approaches a local optimum of
the BMI problem via solving a series of convex optimization problems. This
algorithm is incorporated in a branch-and-bound framework that searches for the
global optimum in a divide-and-conquer fashion. We present a weak completeness
result of our method, namely, a barrier certificate is guaranteed to be found
(under some mild assumptions) whenever there exists an inductive invariant (in
the form of a given template) that suffices to certify safety of the system.
Experimental results on benchmarks demonstrate the effectiveness and efficiency
of our approach.Comment: To be published in Inf. Comput. arXiv admin note: substantial text
overlap with arXiv:2105.1431