Copositivity and constrained fractional quadratic programs

Abstract We provide Completely Positive and Copositive Optimization formulations for the Constrained Fractional Quadratic Problem (CFQP) and Standard Fractional Quadratic Problem (StFQP). Based on these formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which can be used in a global optimization branch-and-bound approach. Applications of the CFQP and StFQP, related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed

Canonical Duality Theory for Global Optimization problems and applications

The canonical duality theory is studied, through a discussion on a general global optimization problem and applications on fundamentally important problems. This general problem is a formulation of the minimization problem with inequality constraints, where the objective function and constraints are any convex or nonconvex functions satisfying certain decomposition conditions. It covers convex problems, mixed integer programming problems and many other nonlinear programming problems. The three main parts of the canonical duality theory are canonical dual transformation, complementary-dual principle and triality theory. The complementary-dual principle is further developed, which conventionally states that each critical point of the canonical dual problem is corresponding to a KKT point of the primal problem with their sharing the same function value. The new result emphasizes that there exists a one-to-one correspondence between KKT points of the dual problem and of the primal problem and each pair of the corresponding KKT points share the same function value, which implies that there is truly no duality gap between the canonical dual problem and the primal problem. The triality theory reveals insightful information about global and local solutions. It is shown that as long as the global optimality condition holds true, the primal problem is equivalent to a convex problem in the dual space, which can be solved efficiently by existing convex methods; even if the condition does not hold, the convex problem still provides a lower bound that is at least as good as that by the Lagrangian relaxation method. It is also shown that through examining the canonical dual problem, the hidden convexity of the primal problem is easily observable. The canonical duality theory is then applied to dealing with three fundamentally important problems. The first one is the spherically constrained quadratic problem, also referred to as the trust region subproblem. The canonical dual problem is onedimensional and it is proved that the primal problem, no matter with convex or nonconvex objective function, is equivalent to a convex problem in the dual space. Moreover, conditions are found which comprise the boundary that separates instances into “hard case” and “easy case”. A canonical primal-dual algorithm is developed, which is able to efficiently solve the problem, including the “hard case”, and can be used as a unified method for similar problems. The second one is the binary quadratic problem, a fundamental problem in discrete optimization. The discussion is focused on lower bounds and analytically solvable cases, which are obtained by analyzing the canonical dual problem with perturbation techniques. The third one is a general nonconvex problem with log-sum-exp functions and quartic polynomials. It arises widely in engineering science and it can be used to approximate nonsmooth optimization problems. The work shows that problems can still be efficiently solved, via the canonical duality approach, even if they are nonconvex and nonsmooth.Doctor of Philosoph

Extremum-Seeking Guidance and Conic-Sector-Based Control of Aerospace Systems

This dissertation studies guidance and control of aerospace systems. Guidance algorithms are used to determine desired trajectories of systems, and in particular, this dissertation examines constrained extremum-seeking guidance. This type of guidance is part of a class of algorithms that drives a system to the maximum or minimum of a performance function, where the exact relation between the function's input and output is unknown. This dissertation abstracts the problem of extremum-seeking to constrained matrix manifolds. Working with a constrained matrix manifold necessitates mathematics other than the familiar tools of linear systems. The performance function is optimized on the manifold by estimating a gradient using a Kalman filter, which can be modified to accommodate a wide variety of constraints and can filter measurement noise. A gradient-based optimization technique is then used to determine the extremum of the performance function. The developed algorithms are applied to aircraft and spacecraft. Control algorithms determine which system inputs are required to drive the systems outputs to follow the trajectory given by guidance. Aerospace systems are typically nonlinear, which makes control more challenging. One approach to control nonlinear systems is linear parameter varying (LPV) control, where well-established linear control techniques are extended to nonlinear systems. Although LPV control techniques work quite well, they require an LPV model of a system. This model is often an approximation of the real nonlinear system to be controlled, and any stability and performance guarantees that are derived using the system approximation are usually void on the real system. A solution to this problem can be found using the Passivity Theorem and the Conic Sector Theorem, two input-output stability theories, to synthesize LPV controllers. These controllers guarantee closed-loop stability even in the presence of system approximation. Several control techniques are derived and implemented in simulation and experimentation, where it is shown that these new controllers are robust to plant uncertainty.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/143993/1/aexwalsh_1.pd

Advances in combined architecture, plant, and control design

The advancement of many engineering systems relies on novel design methodologies, design formulations, design representations, and other advancements. In this dissertation, we consider three broad design domains: architecture, plant, and control. These domains cover most of the potential design decision elements in an actively-controlled engineering system. In this dissertation, strategic aspects of this combined problem are addressed. The task of representing and generating candidate architectures is addressed with methods developed based on colored graphs built by enumerating all perfect matchings of a specified catalog of components. The proposed approach captures all architectures under specific assumptions. General combined plant and control design (or co-design) problems are examined. Previous work in co-design theory imposed restrictions on the type of problems that could be posed. Here many of those restrictions are lifted. The problem formulations and optimality conditions for both the simultaneous and nested solution strategies are given along with a detailed discussion of the two methods. Direct transcription is also discussed as it enables the solution of general co-design problems by approximating the problem. Motivated primarily by the need for efficient methods to solve certain control problems that emerge using the nested co-design method, an automated problem generation procedure is developed to support easy specification of linear-quadratic dynamic optimization problems using direct transcription and quadratic programming. Pseudospectral and single-step methods (including the zero-order hold) are all implemented in this unified framework and comparisons are made. Three detailed engineering design case studies are presented. The results from the enumeration and evaluation of all passive analog circuits with up to a certain number of components are used to synthesize low-pass filters and circuits that match a certain magnitude response. Advantages and limitations of enumerative approaches are highlighted in this case study, along with comparisons to circuits synthesized via evolutionary computation; many similarities are found in the topologies. The second case study tackles a complex co-design problem with the design of strain-actuated solar arrays for spacecraft precision pointing and jitter reduction. Nested co-design is utilized along with a linear-quadratic inner loop problem to obtain solutions efficiently. A simpler, scaled problem is analyzed to gain general insights into these results. This is accomplished with a unified theory of scaling in dynamic optimization. The final case study involves the design of active vehicle suspensions. All three design domains are considered in this problem. A class of architecture, plant, and control design problems which utilize linear physical elements is discussed. This problem class can be solved using the methods in this dissertation

Accelerated Optimization Algorithms for Statistical 3D X-ray Computed Tomography Image Reconstruction.

X-ray computed tomography (CT) has been widely celebrated for its ability to visualize patient anatomy, but increasing radiation exposure to patients is a concern. Statistical image reconstruction algorithms in X-ray CT can provide improved image quality for reduced dose levels in contrast to the conventional filtered back-projection (FBP) methods. However, the statistical approach requires substantial computation time. Therefore, this dissertation focuses on developing fast iterative algorithms for statistical reconstruction. Ordered subsets (OS) methods have been used widely in tomography problems, because they reduce the computational cost by using only a subset of the measurement data per iteration. They are already used in commercial PET and SPECT products. However, OS methods require too long a reconstruction time in X-ray CT to be used routinely for every clinical CT scan. In this dissertation, two main approaches are proposed for accelerating OS algorithms, one that uses new optimization transfer approaches and one that combines with momentum algorithms. The first, the separable quadratic surrogates (SQS) methods, one widely used optimization transfer method with OS methods, have been accelerated in three different ways. Among them, a nonuniform (NU) SQS method encouraging larger step sizes for the voxels that are expected to change more has highly accelerated OS methods. Second, combining OS methods and momentum approaches (OS-momentum) in a way that reuses previous updates with almost negligible increased computation resulted in a very fast convergence rate. This version focused on using widely celebrated Nesterov's momentum methods. OS-momentum algorithms sometimes encountered instability, so diminishing step size rule has been adapted for improving the stability while preserving the fast convergence rate. To further accelerate OS-momentum algorithms, this dissertation proposes novel momentum methods that are twice as fast yet have remarkably simple implementations comparable to Nesterov's methods. In addition to OS-type algorithms, one variant of the block coordinate descent (BCD) algorithm, called Axial BCD (ABCD), is investigated, which is specifically designed for 3D CT geometry. Simulated and real patient 3D CT scans are used to examine the acceleration of the proposed algorithms.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/109007/1/kimdongh_1.pd

Solving Multi-objective Integer Programs using Convex Preference Cones

Esta encuesta tiene dos objetivos: en primer lugar, identificar a los individuos que fueron víctimas de algún tipo de delito y la manera en que ocurrió el mismo. En segundo lugar, medir la eficacia de las distintas autoridades competentes una vez que los individuos denunciaron el delito que sufrieron. Adicionalmente la ENVEI busca indagar las percepciones que los ciudadanos tienen sobre las instituciones de justicia y el estado de derecho en Méxic