A Feasible Method for Optimization with Orthogonality Constraints

Minimization with orthogonality constraints (e.g., X'X = I) and/or spherical constraints (e.g., ||x||_2 = 1) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we propose to use a Crank-Nicholson-like update scheme to preserve the constraints and based on it, develop curvilinear search algorithms with lower per-iteration cost compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842% to the best known solution on the largest problem "256c" in QAPLIB can be reached in 5 minutes on a typical laptop

Use of the NLPQLP Sequential Quadratic Programming Algorithm to Solve Rotorcraft Aeromechanical Constrained Optimisation Problems

Optimization of the control vector, configuration and aerodynamic surface design potentially offers significant performance enhancement to rotorcraft systems. These analyses indicated that non-linear programming methods that solve a sequence of related quadratic-programming sub-problems could be used successfully to solve these problems. Accordingly, a license for one of the latest versions of Professor Schittkowski's very successful Sequential Quadratic Programming NLPQLP software was obtained and used to experiment and analyze typical optimization problems of the type encountered in various rotorcraft wind tunnel and flight tests. Emphasis was directed toward obtaining efficiency, robustness and speed in computation

Destruction of Invariant Tori in Volume-Preserving Maps

Invariant rotational tori play an important role in the dynamics of volume-preserving maps. When integrable, all orbits lie on these tori and KAM theory guarantees the persistence of some tori upon perturbation. When these tori have codimension-one they act as boundaries to transport, and therefore play a prominent role in the global stability of the system. For the area-preserving case, Greene's residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits. Even though KAM theory applies to the three-dimensional case, the robustness of tori in such systems is still poorly understood. This dissertation begins by extending Greene's residue criterion to three-dimensional, reversible, volume-preserving maps. The application of Greene's residue criterion requires the repeated computation of periodic orbits, which is costly if the system is nonreversible. We describe a quasi-Newton, Fourier-based scheme to numerically compute the conjugacy of a torus and demonstrate how the growth of the Sobolev norm or singular values of this conjugacy can be used to predict criticality. We will then use this method to study both reversible and nonreversible volume-preserving maps in two and three dimensions. The near-critical conjugacies, and the gaps that form within them, will be explored in the context of Aubry-Mather and Anti-Integrability theory, when applicable. This dissertation will conclude by exploring the locally and globally most robust tori in area-preserving maps

Use of the NLP10x10 Sequential Quadratic Programming Algorithm To Solve Rotorcraft Hub Loads Minimisation Problems

Previous research and experimentation on the use of a non-linear programming constrained optimisation technique to define an optimal control vector for rotorcraft applications indicated that use of this methodology was feasible and desirable in many cases. In particular, use of non-linear programming methods that solve a sequence of related quadratic-programming sub-problems were used successfully to solve these problems. Accordingly, a licence for one of the latest versions of Professor Klaus Schittkowskis very successful Sequential Quadratic Programming NLPQLP software was obtained and used to experiment with and analyse typical optimisation problems of the type encountered in various rotorcraft wind tunnel and flight tests. This research resulted in the development of the general NLPQLP Computation System that could be used to solve problems of the type encountered in various rotorcraft applications where there is a linear dependence of the measurement vector on the control vector, and where equality andor inequality constraints might be imposed. This development was accomplished on a mainframe computer not part of actual wind tunnel andor flight-test experiment, but in a format which was transferable to wind tunnel lap-top computers. Emphasis was directed toward obtaining efficiency, robustness and speed in computation.The System was developed in support of the five-bladed SMART Rotor Active Flap Rotor Hub Loads analytical minimisation research. The design and development of the Computation System was tailored to address the particular requirements of the problem to minimise a performance metric function of measured hub load harmonic angular couple components by optimising the control vector harmonic flap angular couple components subject to constraints on the amplitudes of these control vector harmonic flap angular couple components. In addition, to facilitate real time wind tunnel experimentation, the ability to rapidly selectchange the particular hub load harmonic angular couple components andor the particular control vector harmonic angular couple components to be considered in the optimisation procedure was provided in the System. This capability allows the singling out of particular hub load frequencies andor particular flap angle frequencies to be analysed during testing operations. The System was used very successfully for the SMART Active Flap Rotor Hub minimisation problems considered in the study, the results of which were presented at the American Helicopter Society Fifth Decennial Aeromechanics Specialist Conference in January 2014. Excellent agreement between cases initiated with best guess starting estimates for the control vector elements and cases initiated with zero control vector starting element estimates resulted, indicating the robustness of the NLP10x10 algorithm