Merit functions: a bridge between optimization and equilibria

In the last decades, many problems involving equilibria, arising from engineering, physics and economics, have been formulated as variational mathematical models. In turn, these models can be reformulated as optimization problems through merit functions. This paper aims at reviewing the literature about merit functions for variational inequalities, quasi-variational inequalities and abstract equilibrium problems. Smoothness and convexity properties of merit functions and solution methods based on them will be presented

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Nonmonotone local minimax methods for finding multiple saddle points

In this paper, by designing a normalized nonmonotone search strategy with the Barzilai--Borwein-type step-size, a novel local minimax method (LMM), which is a globally convergent iterative method, is proposed and analyzed to find multiple (unstable) saddle points of nonconvex functionals in Hilbert spaces. Compared to traditional LMMs with monotone search strategies, this approach, which does not require strict decrease of the objective functional value at each iterative step, is observed to converge faster with less computations. Firstly, based on a normalized iterative scheme coupled with a local peak selection that pulls the iterative point back onto the solution submanifold, by generalizing the Zhang--Hager (ZH) search strategy in the optimization theory to the LMM framework, a kind of normalized ZH-type nonmonotone step-size search strategy is introduced, and then a novel nonmonotone LMM is constructed. Its feasibility and global convergence results are rigorously carried out under the relaxation of the monotonicity for the functional at the iterative sequences. Secondly, in order to speed up the convergence of the nonmonotone LMM, a globally convergent Barzilai--Borwein-type LMM (GBBLMM) is presented by explicitly constructing the Barzilai--Borwein-type step-size as a trial step-size of the normalized ZH-type nonmonotone step-size search strategy in each iteration. Finally, the GBBLMM algorithm is implemented to find multiple unstable solutions of two classes of semilinear elliptic boundary value problems with variational structures: one is the semilinear elliptic equations with the homogeneous Dirichlet boundary condition and another is the linear elliptic equations with semilinear Neumann boundary conditions. Extensive numerical results indicate that our approach is very effective and speeds up the LMMs significantly.Comment: 32 pages, 7 figures; Accepted by Journal of Computational Mathematics on January 3, 202

A vision-based optical character recognition system for real-time identification of tractors in a port container terminal

Automation has been seen as a promising solution to increase the productivity of modern sea port container terminals. The potential of increase in throughput, work efficiency and reduction of labor cost have lured stick holders to strive for the introduction of automation in the overall terminal operation. A specific container handling process that is readily amenable to automation is the deployment and control of gantry cranes in the container yard of a container terminal where typical operations of truck identification, loading and unloading containers, and job management are primarily performed manually in a typical terminal. To facilitate the overall automation of the gantry crane operation, we devised an approach for the real-time identification of tractors through the recognition of the corresponding number plates that are located on top of the tractor cabin. With this crucial piece of information, remote or automated yard operations can then be performed. A machine vision-based system is introduced whereby these number plates are read and identified in real-time while the tractors are operating in the terminal. In this paper, we present the design and implementation of the system and highlight the major difficulties encountered including the recognition of character information printed on the number plates due to poor image integrity. Working solutions are proposed to address these problems which are incorporated in the overall identification system.postprin

Understanding Modern Techniques in Optimization: Frank-Wolfe, Nesterov's Momentum, and Polyak's Momentum

In the first part of this dissertation research, we develop a modular framework that can serve as a recipe for constructing and analyzing iterative algorithms for convex optimization. Specifically, our work casts optimization as iteratively playing a two-player zero-sum game. Many existing optimization algorithms including Frank-Wolfe and Nesterov's acceleration methods can be recovered from the game by pitting two online learners with appropriate strategies against each other. Furthermore, the sum of the weighted average regrets of the players in the game implies the convergence rate. As a result, our approach provides simple alternative proofs to these algorithms. Moreover, we demonstrate that our approach of optimization as iteratively playing a game leads to three new fast Frank-Wolfe-like algorithms for some constraint sets, which further shows that our framework is indeed generic, modular, and easy-to-use. In the second part, we develop a modular analysis of provable acceleration via Polyak's momentum for certain problems, which include solving the classical strongly quadratic convex problems, training a wide ReLU network under the neural tangent kernel regime, and training a deep linear network with an orthogonal initialization. We develop a meta theorem and show that when applying Polyak's momentum for these problems, the induced dynamics exhibit a form where we can directly apply our meta theorem. In the last part of the dissertation, we show another advantage of the use of Polyak's momentum -- it facilitates fast saddle point escape in smooth non-convex optimization. This result, together with those of the second part, sheds new light on Polyak's momentum in modern non-convex optimization and deep learning.Comment: PhD dissertation at Georgia Tech. arXiv admin note: text overlap with arXiv:2010.0161

Provably effective algorithms for min-max optimization

Many fundamental machine learning tasks can be formulated as min-max optimization. This motivates us to design effective and efficient first-order methods that provably converge to the global min-max points. For this purpose, this thesis focuses on designing practical algorithms for several specific machine learning tasks. We considered some different settings: unconstrained or constrained strongly-convex (strongly-)concave, constrained convex-concave, and nonconvex-concave problems. We tackle the following concrete questions by studying the above problems: 1. Can we reformulate a single minimization problem to two-player games to help reduce the computational complexity of finding global optimal points? 2. Can projection-free algorithms achieve last-iterate convergence for constrained min-max optimization problems with the convex-concave landscape? 3. Can we show that stochastic gradient descent-ascent, a method commonly used in practice for GAN training, actually finds global optima and can learn a target distribution? We make progress on these questions by proposing practical algorithms with theoretical guarantees. We also present extensive empirical studies to verify the effectiveness of our proposed methods.Computational Science, Engineering, and Mathematic

Job shop scheduling with artificial immune systems

The job shop scheduling is complex due to the dynamic environment. When the information of the jobs and machines are pre-defined and no unexpected events occur, the job shop is static. However, the real scheduling environment is always dynamic due to the constantly changing information and different uncertainties. This study discusses this complex job shop scheduling environment, and applies the AIS theory and switching strategy that changes the sequencing approach to the dispatching approach by taking into account the system status to solve this problem. AIS is a biological inspired computational paradigm that simulates the mechanisms of the biological immune system. Therefore, AIS presents appealing features of immune system that make AIS unique from other evolutionary intelligent algorithm, such as self-learning, long-lasting memory, cross reactive response, discrimination of self from non-self, fault tolerance, and strong adaptability to the environment. These features of AIS are successfully used in this study to solve the job shop scheduling problem. When the job shop environment is static, sequencing approach based on the clonal selection theory and immune network theory of AIS is applied. This approach achieves great performance, especially for small size problems in terms of computation time. The feature of long-lasting memory is demonstrated to be able to accelerate the convergence rate of the algorithm and reduce the computation time. When some unexpected events occasionally arrive at the job shop and disrupt the static environment, an extended deterministic dendritic cell algorithm (DCA) based on the DCA theory of AIS is proposed to arrange the rescheduling process to balance the efficiency and stability of the system. When the disturbances continuously occur, such as the continuous jobs arrival, the sequencing approach is changed to the dispatching approach that involves the priority dispatching rules (PDRs). The immune network theory of AIS is applied to propose an idiotypic network model of PDRs to arrange the application of various dispatching rules. The experiments show that the proposed network model presents strong adaptability to the dynamic job shop scheduling environment.postprin

Learning-based Optimization for Signal and Image Processing

Incorporating machine learning techniques into optimization problems and solvers attracts increasing attention. Given a particular type of optimization problem that needs to be solved repeatedly, machine learning techniques can find some features for this category of optimization and develop algorithms with excellent performance. This thesis deals with algorithms and convergence analysis in learning-based optimization in three aspects: learning dictionaries, learning optimization solvers and learning regularizers.Learning dictionaries for sparse coding is significant for signal processing. Convolutional sparse coding is a form of sparse coding with a structured, translation invariant dictionary. Most convolutional dictionary learning algorithms to date operate in the batch mode, requiring simultaneous access to all training images during the learning process, which results in very high memory usage, and severely limits the training data size that can be used. I proposed two online convolutional dictionary learning algorithms that offered far better scaling of memory and computational cost than batch methods and provided a rigorous theoretical analysis of these methods.Learning fast solvers for optimization is a rising research topic. In recent years, unfolding iterative algorithms as neural networks has become an empirical success in solving sparse recovery problems. However, its theoretical understanding is still immature, which prevents us from fully utilizing the power of neural networks. I studied unfolded ISTA (Iterative Shrinkage Thresholding Algorithm) for sparse signal recovery and established its convergence. Based on the properties of parameters required by convergence, the model can be significantly simplified and, consequently, has much less training cost and better recovery performance.Learning regularizers or priors improves the performance of optimization solvers, especially for signal and image processing tasks. Plug-and-play (PnP) is a non-convex framework that integrates modern priors, such as BM3D or deep learning-based denoisers, into ADMM or other proximal algorithms. Although PnP has been recently studied extensively with great empirical success, theoretical analysis addressing even the most basic question of convergence has been insufficient. In this thesis, the theoretical convergence of PnP-FBS and PnP-ADMM was established, without using diminishing stepsizes, under a certain Lipschitz condition on the denoisers. Furthermore, real spectral normalization was proposed for training deep learning-based denoisers to satisfy the proposed Lipschitz condition