Second-Order Optimality Conditions for Nonconvex Set-Constrained Optimization Problems

In this paper, we study second-order optimality conditions for nonconvex set-constrained optimization problems. For a convex set-constrained optimization problem, it is well known that second-order optimality conditions involve the support function of the second-order tangent set. In this paper, we propose two approaches for establishing second-order optimality conditions for the nonconvex case. In the first approach, we extend the concept of the support function so that it is applicable to general nonconvex set-constrained problems, whereas in the second approach, we introduce the notion of the directional regular tangent cone and apply classical results of convex duality theory. Besides the second-order optimality conditions, the novelty of our approach lies in the systematic introduction and use, respectively, of directional versions of well-known concepts from variational analysis.

LMBOPT: a limited memory method for bound-constrained optimization

Recently, Neumaier and Azmi gave a comprehensive convergence theory for a generic algorithm for bound constrained optimization problems with a continuously differentiable objective function. The algorithm combines an active set strategy with a gradient-free line search along a piecewise linear search path defined by directions chosen to reduce zigzagging. This paper describes , an efficient implementation of this scheme. It employs new limited memory techniques for computing the search directions, improves by adding various safeguards relevant when finite precision arithmetic is used, and adds many practical enhancements in other details. The paper compares and several other solvers on the unconstrained and bound constrained problems from the collection and makes recommendations on which solver to use and when. Depending on the problem class, the problem dimension, and the precise goal, the best solvers are , , and .

The Isotropic Material Design of In-Plane Loaded Elasto-Plastic Plates

This paper puts forward a new version of the Isotropic Material Design method for the optimum design of structures made of an elasto-plastic material within the Hencky-Nadai-Ilyushin theory. This method provides the optimal layouts of the moduli of isotropy to make the overall compliance minimal. Thus, the bulk and shear moduli are the only design variables, both assumed as non-negative fields. The trace of the Hooke tensor represents the unit cost of the design. The yield condition is assumed to be independent of the design variables, to make the design process as simple as possible. By eliminating the design variables, the optimum design problem is reduced to the pair of the two mutually dual Linear Constrained Problems (LCP). The solution to the LCP stress-based problem directly determines the layout of the optimal moduli. A numerical method has been developed to construct approximate solutions, which paves the way for constructing the final layouts of the elastic moduli. Selected illustrative solutions are reported, corresponding to various data concerning the yield limit and the cost of the design. The yield condition introduced in this paper results in bounding the values of the optimal moduli in the places of possible stress concentration, such as reentrant corners.

Distributed Processing of Blind Source Separation

<p>Communication is performed by transmitting signals through a medium. It is common that signals originating from different sources are mixed in the transport medium. The operation of separating source signals without prior information about the sources is referred to as blind source separation (BSS). Blind source separation for wireless sensor networks has recently received attention because of low cost and the easy coverage of large areas. Distributed processing is attractive as it is scalable and consumes low power. Existing distributed BSS algorithms either require a fully connected pattern of connectivity, to ensure the good performance, or require a high computational load at each sensor node, to enhance the scalability. This motivates us to develop distributed BSS algorithms that can be implemented over any arbitrary graph with fully shared computations and with good performance. This thesis presents three studies on distributed algorithms. The first two studies are on existing distributed algorithms that are used in linearly constrained convex optimization problems, which are common in signal processing and machine learning. The studies are aimed at improving the algorithms in terms of computational complexity, communication cost, processors coordination and scalability. This makes them more suitable for implementation on sensor networks, thus forming a basis for the development of distributed BSS algorithms on sensor networks in our third study. In the first study, we consider constrained problems in which the constraint includes a weighted sum of all the decision variables. By formulating a constrained dual problem associated to the original constrained problem, we were able to develop a distributed algorithm that can be run both synchronously and asynchronously on any arbitrary graph with lower communication cost than traditional distributed algorithms. In the second study, we consider constrained problems in which the constraint is separable. By making use of the augmented Lagrangian function and splitting the dual variable (Lagrange multiplier) associated to each partial constraint, we were able to develop a distributed fully asynchronous algorithm with lower computational complexity than traditional distributed algorithms. The simplicity of the algorithm is the consequence of approximating the constraint on the equality of the decoupled dual variables. We also provide a measure of the inaccuracy in such an approximation on the optimal value of the primal objective function. Finally, in the third study, we investigate distributed processing solutions for BSS on sensor networks. We propose two distributed processing schemes for BSS that we refer to as scheme 1 and scheme 2. In scheme 1, each sensor node estimates one specific source signal while in scheme 2, by formulating a consensus optimization problem, each sensor node estimates all source signals in a fully shared computation manner. Our proposed algorithms carry the following features: low computational complexity, low power consumption, low data transmission rate, scalability and excellent performance over arbitrary graphs. Although all of our proposed algorithms share the aforementioned properties, each of them is superior in one or some of the features compared to the others. Comparative experimental results show that among all our proposed distributed BSS algorithms, a variant of scheme 1 performs best when all features are considered. This is achieved by making use of the concept of pairwise mutual information along with adding a sparsity assumption on the parameters of the model that is used in BSS.</p>

Distributed Processing of Blind Source Separation

<p>Communication is performed by transmitting signals through a medium. It is common that signals originating from different sources are mixed in the transport medium. The operation of separating source signals without prior information about the sources is referred to as blind source separation (BSS). Blind source separation for wireless sensor networks has recently received attention because of low cost and the easy coverage of large areas. Distributed processing is attractive as it is scalable and consumes low power. Existing distributed BSS algorithms either require a fully connected pattern of connectivity, to ensure the good performance, or require a high computational load at each sensor node, to enhance the scalability. This motivates us to develop distributed BSS algorithms that can be implemented over any arbitrary graph with fully shared computations and with good performance. This thesis presents three studies on distributed algorithms. The first two studies are on existing distributed algorithms that are used in linearly constrained convex optimization problems, which are common in signal processing and machine learning. The studies are aimed at improving the algorithms in terms of computational complexity, communication cost, processors coordination and scalability. This makes them more suitable for implementation on sensor networks, thus forming a basis for the development of distributed BSS algorithms on sensor networks in our third study. In the first study, we consider constrained problems in which the constraint includes a weighted sum of all the decision variables. By formulating a constrained dual problem associated to the original constrained problem, we were able to develop a distributed algorithm that can be run both synchronously and asynchronously on any arbitrary graph with lower communication cost than traditional distributed algorithms. In the second study, we consider constrained problems in which the constraint is separable. By making use of the augmented Lagrangian function and splitting the dual variable (Lagrange multiplier) associated to each partial constraint, we were able to develop a distributed fully asynchronous algorithm with lower computational complexity than traditional distributed algorithms. The simplicity of the algorithm is the consequence of approximating the constraint on the equality of the decoupled dual variables. We also provide a measure of the inaccuracy in such an approximation on the optimal value of the primal objective function. Finally, in the third study, we investigate distributed processing solutions for BSS on sensor networks. We propose two distributed processing schemes for BSS that we refer to as scheme 1 and scheme 2. In scheme 1, each sensor node estimates one specific source signal while in scheme 2, by formulating a consensus optimization problem, each sensor node estimates all source signals in a fully shared computation manner. Our proposed algorithms carry the following features: low computational complexity, low power consumption, low data transmission rate, scalability and excellent performance over arbitrary graphs. Although all of our proposed algorithms share the aforementioned properties, each of them is superior in one or some of the features compared to the others. Comparative experimental results show that among all our proposed distributed BSS algorithms, a variant of scheme 1 performs best when all features are considered. This is achieved by making use of the concept of pairwise mutual information along with adding a sparsity assumption on the parameters of the model that is used in BSS.</p>

The Incentive Mechanism of Knowledge Sharing in Cross-Border Business Models Based on Digital Technologies

This paper aims to solve the time-constrained problems of knowledge sharing caused by geographical distance and cultural differences in cross-border business models by proposing a novel knowledge sharing model based on principal–agent theory. Given that digital technologies (DTs) can solve the information asymmetry issue, this paper analyses and compares the contract parameters given by the principal, the efforts of the agent, and the changes in the expected profits of both parties before and after the application of DTs and therefore discusses the influence of various relevant factors in incentive contracts; the relationship between the expected profit of both parties and the various relevant factors is analyzed through numerical simulations. The results show that, in cross-border business models considering the time value of knowledge, the principal is affected not only by “information rent” and “channel loss” but also by the “time cost”. The application of DTs can effectively reduce all three of these costs. More importantly, the principal’s incentive coefficient and the agent’s effort are related to this time constraint and the application of DTs.

Epidemic Control and Resource Allocation: Approaches and Implications for the Management of COVID-19

The experience with COVID-19 underscores a classic public policy choice problem: how should policymakers determine how to allocate constrained budgets, limited equipment, under-resourced hospitals and stretched personnel to limit the spread of the virus. This article presents an overview of the general literature on resource allocation in epidemics and assess how it informs our understanding of COVID-19. We highlight the peculiarities of the pandemic that call for a rethinking of existing approaches to resource allocation. In particular, we analyse how the experience of COVID-19 informs our understanding and modelling of the optimal resource allocation problem in epidemics. Our delineation of the literature focuses on resource constraint as the key variable. A qualitative appraisal indicates that the current suit of models for understanding the resource allocation problem requires adaptations to advance our management of COVID-19 or similar future epidemics. Particularly under-studied areas include issues of uncertainty, potential for co-epidemics, the role of global connectivity, and resource constrained problems arising from depressed economic activity. Incorporating various global dimensions of COVID-19 into resource allocation modelling such a centralized versus decentralized resource control and the role of geostrategic interests could yield crucial insights. This will require multi-disciplinary approaches to the resource allocation problem. JEL Classifications: I14, I18, E61, D60, H4, H12

The Complexity of the Ideal Membership Problem for Constrained Problems Over the Boolean Domain

Given an ideal I and a polynomial f the Ideal Membership Problem (IMP) is to test if f ϵ I . This problem is a fundamental algorithmic problem with important applications and notoriously intractable. We study the complexity of the IMP for combinatorial ideals that arise from constrained problems over the Boolean domain. As our main result, we identify the borderline of tractability. By using Gröbner bases techniques, we extend Schaefer’s dichotomy theorem [STOC, 1978] which classifies all Constraint Satisfaction Problems (CSPs) over the Boolean domain to be either in P or NP-hard. Moreover, our result implies necessary and sufficient conditions for the efficient computation of Theta Body Semi-Definite Programming (SDP) relaxations, identifying therefore the borderline of tractability for constraint language problems. This article is motivated by the pursuit of understanding the recently raised issue of bit complexity of Sum-of-Squares (SoS) proofs [O’Donnell, ITCS, 2017]. Raghavendra and Weitz [ICALP, 2017] show how the IMP tractability for combinatorial ideals implies bounded coefficients in SoS proofs.