No abstract

A method is proposed for solving equality constrained nonlinear optimization problems involving twice continuously differentiable functions. The method employs a trust funnel approach consisting of two phases: a first phase to locate an ǫ-feasible point and a second phase to seek optimality while maintaining at least ǫ-feasibility. A two-phase approach of this kind based on a cubic regularization methodology was recently proposed along with a supporting worst-case iteration complexity analysis. Unfortunately, however, in that approach, the objective function is completely ignored in the first phase when ǫ-feasibility is sought. The main contribution of the method proposed in this paper is that the same worst-case iteration complexity is achieved, but with a first phase that also accounts for improvements in the objective function. As such, the method typically requires fewer iterations in the second phase, as the results of numerical experiments demonstrate.1 Unfortunately, however, the method in [8] represents a departure from the current state-of-the-art SQP methods that offer the best practical performance. One of the main reasons for this is that contemporary SQP methods seek feasibility and optimality simultaneously. By contrast, one of the main reasons that the approach from [8] does not offer practical benefits is that the first phase of the algorithm entirely ignores the objective function, meaning that numerous iterations might need to be performed before the objective function influences the trajectory of the algorithm.The algorithm proposed in this paper can be considered a next step in the design of practical algorithms for equality constrained optimization with good worst-case iteration complexity properties. Ours is also a two-phase approach, but is closer to the SQP-type methods representing the state-of-the-art for solving equality constrained problems. In particular, the first phase of our proposed approach follows a trust funnel methodology that locates an ǫ-feasible point in O(ǫ −3/2 ) iterations while also attempting to yield improvements in the objective function. Borrowing ideas from the trust region method known as trace [15], we prove that our method attains the same worst-case iteration complexity bounds as those offered by [8], and show with numerical experiments that consideration of the objective function in the first phase typically results in the second phase requiring fewer iterations.1.1. Organization. In the remainder of this section, we introduce notation that is used throughout the remainder of the paper and cover preliminary material on equality constrained nonlinear optimization. In §2, we motivate and describe our proposed "phase 1" method for locating an ǫ-feasible point while also attempting to reduce the objective function. An analysis of the convergence and worst-case iteration complexity of this phase 1 method is presented in §3. Strategies and corresponding convergence/complexity guarantees for "phase 2" are the subject of §4, the results of numerical experiments are pr...

An algorithm for solving smooth nonconvex optimization problems is proposed that, in the worst-case, takes O(ε −3/2 ) iterations to drive the norm of the gradient of the objective function below a prescribed positive real number ε and can take O(ε −3 ) iterations to drive the leftmost eigenvalue of the Hessian of the objective above −ε. The proposed algorithm is a general framework that covers a wide range of techniques including quadratically and cubically regularized Newton methods, such as the Adaptive Regularisation using Cubics (ARC) method and the recently proposed Trust-Region Algorithm with Contractions and Expansions (TRACE). The generality of our method is achieved through the introduction of generic conditions that each trial step is required to satisfy, which in particular allow for inexact regularized Newton steps to be used. These conditions center around a new subproblem that can be approximately solved to obtain trial steps that satisfy the conditions. A new instance of the framework, distinct from ARC and TRACE, is described that may be viewed as a hybrid between quadratically and cubically regularized Newton methods. Numerical results demonstrate that our hybrid algorithm outperforms a cublicly regularized Newton method.Second, they do not consider second-order convergence or complexity properties, although they might be able to do so by incorporating second-order conditions similar to ours. Third, they focus on strategies for identifying an appropriate value for the regularization parameter. An implementation of our method might consider their proposals, but could employ other strategies as well. In any case, overall, we believe that our papers are quite distinct, and in some ways are complementary.ORGANIZATION In §2, we present our general framework that is formally stated as Algorithm 1. In §3, we prove that our framework enjoys first-order convergence (see §3.1), an optimal first-order complexity (see §3.2), and certain second-order convergence and complexity guarantees (see §3.3).In §4, we show that ARC and TRACE can be viewed as special cases of our framework, and present yet another instance that is distinct from these methods. In §5, we present details of implementations of a cubic regularization method and our newly proposed instance of our framework, and provide the results of numerical experiments with both. Finally, in §6, we present final comments.NOTATION We use R + to denote the set of nonnegative scalars, R ++ to denote the set of positive scalars, and N + to denote the set of nonnegative integers. Given a real symmetric matrix A, we write A 0 (respectively, A 0) to indicate that A is positive semidefinite (respectively, positive definite). Given a pair of scalars (a, b) ∈ R × R, we write a ⊥ b to indicate that ab = 0. Similarly, given such a pair, we denote their maximum as max{a, b} and their minimum as min{a, b}. Given a vector v, we denote its (Euclidean) 2 -norm as v . Finally, given a discrete set S , we denote its cardinality by |S |.

No abstract

With the growing prevalence of obesity and the public health implications, it is critical to develop and evaluate potential interventions. One approach is to investigate the spread of positive health outcomes through a social network. We employ the ground concepts of spread maximization problem and adapt it to best reflect the dynamics of a weight loss intervention. A diffusion model is then employed for the propagation of weight loss effect throughout the network of obese and overweight individuals. This diffusion model integrates both personal attributes and network-related attributes of an individual while engaged in a weight loss program. Simulation findings suggest that choosing initial agents based on individual attributes, including ability to lose weight, body mass index, and threshold, produced the highest total weight loss in the network. Greedy algorithm was also applied to choose the most effective subset of initial seeds.