Application of Water Cycle algorithm to Stochastic Fractional Programming Problem

This paper presents an application of Water Cycle algorithm (WCA) in solving stochastic programming problems. In particular, Linear stochastic fractional programming problems are considered which are solved by WCA and solutions are compared with Particle Swarm Optimization, Differential Evolution, and Whale Optimization Algorithm and the results from literature. The constraints are handled by converting constrained optimization problem into an unconstrained optimization problem using Augmented Lagrangian Method. Further, a fractional stochastic transportation problem is examined as an application of the stochastic fractional programming problem. In terms of efficiency of algorithms and the ability to find optimal solutions, WCA gives highly significant results in comparison with the other metaheuristic algorithms and the quoted results in the literature which demonstrates that WCA algorithm has 100% convergence in all the problems. Moreover, non-parametric hypothesis tests are performed and which indicates that WCA presents better results as compare to the other algorithms.

Application of Water Cycle algorithm to Stochastic Fractional Programming Problem

This paper presents an application of Water Cycle algorithm (WCA) in solving stochastic programming problems. In particular, Linear stochastic fractional programming problems are considered which are solved by WCA and solutions are compared with Particle Swarm Optimization, Differential Evolution, and Whale Optimization Algorithm and the results from literature. The constraints are handled by converting constrained optimization problem into an unconstrained optimization problem using Augmented Lagrangian Method. Further, a fractional stochastic transportation problem is examined as an application of the stochastic fractional programming problem. In terms of efficiency of algorithms and the ability to find optimal solutions, WCA gives highly significant results in comparison with the other metaheuristic algorithms and the quoted results in the literature which demonstrates that WCA algorithm has 100% convergence in all the problems. Moreover, non-parametric hypothesis tests are performed and which indicates that WCA presents better results as compare to the other algorithms.

On Degenerate Doubly Nonnegative Projection Problems

The doubly nonnegative (DNN) cone, being the set of all positive semidefinite matrices whose elements are nonnegative, is a popular approximation of the computationally intractable completely positive cone. The major difficulty for implementing a Newton-type method to compute the projection of a given large-scale matrix onto the DNN cone lies in the possible failure of the constraint nondegeneracy, a generalization of the linear independence constraint qualification for nonlinear programming. Such a failure results in the singularity of the Jacobian of the nonsmooth equation representing the Karush–Kuhn–Tucker optimality condition that prevents the semismooth Newton–conjugate gradient method from solving it with a desirable convergence rate. In this paper, we overcome the aforementioned difficulty by solving a sequence of better conditioned nonsmooth equations generated by the augmented Lagrangian method (ALM) instead of solving one aforementioned singular equation. By leveraging the metric subregularity of the normal cone associated with the positive semidefinite cone, we derive sufficient conditions to ensure the dual quadratic growth condition of the underlying problem, which further leads to the asymptotically superlinear convergence of the proposed ALM. Numerical results on difficult randomly generated instances and from the semidefinite programming library are presented to demonstrate the efficiency of the algorithm for computing the DNN projection to a very high accuracy.

On Optimality Conditions for Nonlinear Conic Programming

Sequential optimality conditions play a major role in proving stronger global convergence results of numerical algorithms for nonlinear programming. Several extensions are described in conic contexts, in which many open questions have arisen. In this paper, we present new sequential optimality conditions in the context of a general nonlinear conic framework, which explains and improves several known results for specific cases, such as semidefinite programming, second-order cone programming, and nonlinear programming. In particular, we show that feasible limit points of sequences generated by the augmented Lagrangian method satisfy the so-called approximate gradient projection optimality condition and, under an additional smoothness assumption, the so-called complementary approximate Karush–Kuhn–Tucker condition. The first result was unknown even for nonlinear programming, and the second one was unknown, for instance, for semidefinite programming.

An Efficient Algorithm for Convex Biclustering

We consider biclustering that clusters both samples and features and propose efficient convex biclustering procedures. The convex biclustering algorithm (COBRA) procedure solves twice the standard convex clustering problem that contains a non-differentiable function optimization. We instead convert the original optimization problem to a differentiable one and improve another approach based on the augmented Lagrangian method (ALM). Our proposed method combines the basic procedures in the ALM with the accelerated gradient descent method (Nesterov’s accelerated gradient method), which can attain O(1/k2) convergence rate. It only uses first-order gradient information, and the efficiency is not influenced by the tuning parameter λ so much. This advantage allows users to quickly iterate among the various tuning parameters λ and explore the resulting changes in the biclustering solutions. The numerical experiments demonstrate that our proposed method has high accuracy and is much faster than the currently known algorithms, even for large-scale problems.

Acoustic Room Impulse Response Shaping

<p>Impulse response shaping is a technique for modifying the characteristics of a linear channel to achieve desirable characteristics. The technique is well-known in the field of wireless communication. Acoustic impulse response shaping is used to reduce the effects of reverberation on audio signals propagating inside a room and is thus used for listening room compensation. This thesis addresses innovative approaches for acoustic impulse response shaping. Many techniques have been proposed in the literature for canceling or reducing the effect of reverberation on the audio signal. Impulse response inversion attempts to completely cancel the effect of reverberations whereas impulse response shortening (or shaping) only partly equalizes the room impulse responses. Shortening has less stringent constraints than inversion and this can result in more robust solutions and thus more practically realizable systems. Acoustic impulse response shaping works on measured room impulse responses and designs pre-filters to be placed before the loudspeakers so that the reverberation is reduced at the listening positions. When sampled, the room responses typically contain thousands or tens of thousands (N ) of samples. Thus, the shaping algorithm needs to be computationally fast and memory efficient in order to implement the system in real time. The techniques presented in the literature use interior point methods or steepest descent algorithms, which are computationally slow or require memory of the order of N² . This thesis presents shaping approaches based on the Dual Augmented Lagrangian Method (DALM), known in the literature on sparse reconstruction for its super-linear convergence. The method presented here also makes use of the concept of a Forward Adjoint Oracle (FAO) to make the shaping algorithm memory efficient. Thus, the thesis presents computationally fast and memory efficient shaping algorithms that can be used for practically realizable systems. The thesis also presents robust shaping approaches. The measured room responses may contain measurement errors or noise and can vary from time to time. These variations may be due to changes in atmospheric conditions (such as temperature or humidity) or due to change in position of objects inside a room. While design approaches over multiple microphone positions have been proposed for design of filters that are robust to change in microphone positions, a more rigorous approach is statistical, involving the inclusion of some statistical constraints into the optimization problem. The thesis presents both the approaches viz., a computationally faster version (using DALM) of the already proposed design over multiple positions and a statistically robust shaping formulation. The latter limits the probability of large errors between expected and obtained response to be less than a specified value. This ensures that the solution is robust to variations in the room response. The shaping algorithm works in the time domain, shaping the temporal characteristics of the room response to a desired form. The frequency response of the shaped response can contain potentially undesirable peaks and troughs. This thesis therefore presents an approach for an efficient projection to improve spectral flatness of the resultant response. This algorithm can be combined with the fast and memory efficient DALM based approach to achieve joint time and frequency shaping. Finally, the thesis also presents a computationally fast algorithm based on DALM for pressure matching used in sound field reproduction. Impulse response shaping is applied in sound field reproduction, showing that the levels of pre-reverberation induced by a temperature change can be reduced. This application is different from impulse response shaping approaches presented in the previous chapters and highlights the flexibility of the algorithm developed in this thesis and its wide range of applications.</p>

Acoustic Room Impulse Response Shaping

<p>Impulse response shaping is a technique for modifying the characteristics of a linear channel to achieve desirable characteristics. The technique is well-known in the field of wireless communication. Acoustic impulse response shaping is used to reduce the effects of reverberation on audio signals propagating inside a room and is thus used for listening room compensation. This thesis addresses innovative approaches for acoustic impulse response shaping. Many techniques have been proposed in the literature for canceling or reducing the effect of reverberation on the audio signal. Impulse response inversion attempts to completely cancel the effect of reverberations whereas impulse response shortening (or shaping) only partly equalizes the room impulse responses. Shortening has less stringent constraints than inversion and this can result in more robust solutions and thus more practically realizable systems. Acoustic impulse response shaping works on measured room impulse responses and designs pre-filters to be placed before the loudspeakers so that the reverberation is reduced at the listening positions. When sampled, the room responses typically contain thousands or tens of thousands (N ) of samples. Thus, the shaping algorithm needs to be computationally fast and memory efficient in order to implement the system in real time. The techniques presented in the literature use interior point methods or steepest descent algorithms, which are computationally slow or require memory of the order of N² . This thesis presents shaping approaches based on the Dual Augmented Lagrangian Method (DALM), known in the literature on sparse reconstruction for its super-linear convergence. The method presented here also makes use of the concept of a Forward Adjoint Oracle (FAO) to make the shaping algorithm memory efficient. Thus, the thesis presents computationally fast and memory efficient shaping algorithms that can be used for practically realizable systems. The thesis also presents robust shaping approaches. The measured room responses may contain measurement errors or noise and can vary from time to time. These variations may be due to changes in atmospheric conditions (such as temperature or humidity) or due to change in position of objects inside a room. While design approaches over multiple microphone positions have been proposed for design of filters that are robust to change in microphone positions, a more rigorous approach is statistical, involving the inclusion of some statistical constraints into the optimization problem. The thesis presents both the approaches viz., a computationally faster version (using DALM) of the already proposed design over multiple positions and a statistically robust shaping formulation. The latter limits the probability of large errors between expected and obtained response to be less than a specified value. This ensures that the solution is robust to variations in the room response. The shaping algorithm works in the time domain, shaping the temporal characteristics of the room response to a desired form. The frequency response of the shaped response can contain potentially undesirable peaks and troughs. This thesis therefore presents an approach for an efficient projection to improve spectral flatness of the resultant response. This algorithm can be combined with the fast and memory efficient DALM based approach to achieve joint time and frequency shaping. Finally, the thesis also presents a computationally fast algorithm based on DALM for pressure matching used in sound field reproduction. Impulse response shaping is applied in sound field reproduction, showing that the levels of pre-reverberation induced by a temperature change can be reduced. This application is different from impulse response shaping approaches presented in the previous chapters and highlights the flexibility of the algorithm developed in this thesis and its wide range of applications.</p>

SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning

We study the quadratic cycle cover problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the reduction and the structure of the graph, which is exploited in an efficient algorithm that constructs this matrix for any instance of the problem. To solve our relaxations, we propose an algorithm that incorporates an augmented Lagrangian method into a cutting-plane framework by utilizing Dykstra’s projection algorithm. Our algorithm is suitable for solving SDP relaxations with a large number of cutting-planes. Computational results show that our SDP bounds and efficient cutting-plane algorithm outperform other QCCP bounding approaches from the literature. Finally, we provide several SDP-based upper bounding techniques, among which is a sequential Q-learning method that exploits a solution of our SDP relaxation within a reinforcement learning environment. Summary of Contribution: The quadratic cycle cover problem (QCCP) is the problem of finding a set of node-disjoint cycles covering all the nodes in a graph such that the total interaction cost between successive arcs is minimized. The QCCP has applications in many fields, among which are robotics, transportation, energy distribution networks, and automatic inspection. Besides this, the problem has a high theoretical relevance because of its close connection to the quadratic traveling salesman problem (QTSP). The QTSP has several applications, for example, in bioinformatics, and is considered to be among the most difficult combinatorial optimization problems nowadays. After removing the subtour elimination constraints, the QTSP boils down to the QCCP. Hence, an in-depth study of the QCCP also contributes to the construction of strong bounds for the QTSP. In this paper, we study the application of semidefinite programming (SDP) to obtain strong bounds for the QCCP. Our strongest SDP relaxation is very hard to solve by any SDP solver because of the large number of involved cutting-planes. Because of that, we propose a new approach in which an augmented Lagrangian method is incorporated into a cutting-plane framework by utilizing Dykstra’s projection algorithm. We emphasize an efficient implementation of the method and perform an extensive computational study. This study shows that our method is able to handle a large number of cuts and that the resulting bounds are currently the best QCCP bounds in the literature. We also introduce several upper bounding techniques, among which is a distributed reinforcement learning algorithm that exploits our SDP relaxations.