Well-Posedness and Porosity for Symmetric Optimization Problems

In the present work, we investigate a collection of symmetric minimization problems, which is identified with a complete metric space of lower semi-continuous and bounded from below functions. In our recent paper, we showed that for a generic objective function, the corresponding symmetric optimization problem possesses two solutions. In this paper, we strengthen this result using a porosity notion. We investigate the collection of all functions such that the corresponding optimization problem is well-posed and prove that its complement is a σ-porous set.

A Full Nesterov-Todd Step Infeasible Interior-point Method for Symmetric Optimization in the Wider Neighborhood of the Central Path

In this paper, an improved Interior-Point Method (IPM) for solving symmetric optimization problems is presented. Symmetric optimization (SO) problems are linear optimization problems over symmetric cones. In particular, the method can be efficiently applied to an important instance of SO, a Controlled Tabular Adjustment (CTA) problem which is a method used for Statistical Disclosure Limitation (SDL) of tabular data. The presented method is a full Nesterov-Todd step infeasible IPM for SO. The algorithm converges to ε-approximate solution from any starting point whether feasible or infeasible. Each iteration consists of the feasibility step and several centering steps, however, the iterates are obtained in the wider neighborhood of the central path in comparison to the similar algorithms of this type which is the main improvement of the method. However, the currently best known iteration bound known for infeasible short-step methods is still achieved.

Generic Existence of Solutions of Symmetric Optimization Problems

In this paper we study a class of symmetric optimization problems which is identified with a space of objective functions, equipped with an appropriate complete metric. Using the Baire category approach, we show the existence of a subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every objective function from this intersection the corresponding symmetric optimization problem possesses a solution.

A Novel Learning Rate Schedule in Optimization for Neural Networks and It’s Convergence

The process of machine learning is to find parameters that minimize the cost function constructed by learning the data. This is called optimization and the parameters at that time are called the optimal parameters in neural networks. In the process of finding the optimization, there were attempts to solve the symmetric optimization or initialize the parameters symmetrically. Furthermore, in order to obtain the optimal parameters, the existing methods have used methods in which the learning rate is decreased over the iteration time or is changed according to a certain ratio. These methods are a monotonically decreasing method at a constant rate according to the iteration time. Our idea is to make the learning rate changeable unlike the monotonically decreasing method. We introduce a method to find the optimal parameters which adaptively changes the learning rate according to the value of the cost function. Therefore, when the cost function is optimized, the learning is complete and the optimal parameters are obtained. This paper proves that the method ensures convergence to the optimal parameters. This means that our method achieves a minimum of the cost function (or effective learning). Numerical experiments demonstrate that learning is good effective when using the proposed learning rate schedule in various situations.

Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs

Symmetry in mathematical optimization may create multiple, equivalent solutions. In nonconvex optimization, symmetry can negatively affect algorithm performance, e.g., of branch-and-bound when symmetry induces many equivalent branches. This paper develops detection methods for symmetry groups in quadratically-constrained quadratic optimization problems. Representing the optimization problem with adjacency matrices, we use graph theory to transform the adjacency matrices into binary layered graphs. We enter the binary layered graphs into the software package nauty that generates important symmetric properties of the original problem. Symmetry pattern knowledge motivates a discretization pattern that we use to reduce computation time for an approximation of the point packing problem. This paper highlights the importance of detecting and classifying symmetry and shows that knowledge of this symmetry enables quick approximation of a highly symmetric optimization problem.