Convex Ternary Quartics Are SOS-Convex
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Global Optimization
Strengthening Lasserre’s Hierarchy in Real and Complex Polynomial Optimization
We establish a connection between multiplication operators and shift operators. Moreover, we derive positive semidefinite conditions of finite rank moment sequences and use these conditions to strengthen Lasserre’s hierarchy for real and complex polynomial optimization. Integration of the strengthening technique with sparsity is considered. Extensive numerical experiments show that our strengthening technique can significantly improve … Read more
Revisiting the fitting of the Nelson-Siegel and Svensson models
The Nelson-Siegel and the Svensson models are two of the most widely used models for the term structure of interest rates. Even though the models are quite simple and intuitive, fitting them to market data is numerically challenging and various difficulties have been reported. In this paper, a novel mathematical analysis of the fitting problem … Read more
The best approximation pair problem relative to two subsets in a normed space
In the classical best approximation pair (BAP) problem, one is given two nonempty, closed, convex and disjoint subsets in a finite- or an infinite-dimensional Hilbert space, and the goal is to find a pair of points, each from each subset, which realizes the distance between the subsets. This problem, which has a long history, has … Read more
ε-Optimality in Reverse Optimization
The purpose of this paper is to completely characterize the global approximate optimality (ε-optimality) in reverse convex optimization under the general nonconvex constraint “h(x) ≥ 0”. The main condition presented is obtained in terms of Fenchel’s ε-subdifferentials thanks to El Maghri’s ε-efficiency in difference vector optimization [J. Glob. Optim. 61 (2015) 803–812], after converting the … Read more
Novel stepsize for some accelerated and stochastic optimization methods
New first-order methods now need to be improved to keep up with the constant developments in machine learning and mathematics. They are commonly used methods to solve optimization problems. Among them, the algorithm branch based on gradient descent has developed rapidly with good results achieved. Not out of that trend, in this article, we research … Read more
Exploring Nonlinear Distance Metrics for Lipschitz Constant Estimation in Lower Bound Construction for Global Optimization
Bounds play a crucial role in guiding optimization algorithms, improving their speed and quality, and providing optimality gaps. While Lipschitz constant-based lower bound construction is an effective technique, the quality of the linear bounds depends on the function’s topological properties. In this research, we improve upon this by incorporating nonlinear distance metrics and surrogate approximations … Read more
Refined TSSOS
The moment-sum of squares hierarchy by Lasserre has become an established technique for solving polynomial optimization problems. It provides a monotonically increasing series of tight bounds, but has well-known scalability limitations. For structured optimization problems, the term-sparsity SOS (TSSOS) approach scales much better due to block-diagonal matrices, obtained by completing the connected components of adjacency … Read more
Sparse Polynomial Optimization with Unbounded Sets
This paper considers sparse polynomial optimization with unbounded sets. When the problem possesses correlative sparsity, we propose a sparse homogenized Moment-SOS hierarchy with perturbations to solve it. The new hierarchy introduces one extra auxiliary variable for each variable clique according to the correlative sparsity pattern. Under the running intersection property, we prove that this hierarchy … Read more
Solving moment and polynomial optimization problems on Sobolev spaces
Using standard tools of harmonic analysis, we state and solve the problem of moments for positive measures supported on the unit ball of a Sobolev space of multivariate periodic trigonometric functions. We describe outer and inner semidefinite approximations of the cone of Sobolev moments. They are the basic components of an infinite-dimensional moment-sums of squares … Read more