On the Partial Convexification of the Low-Rank Spectral Optimization: Rank Bounds and Algorithms

A Low-rank Spectral Optimization Problem (LSOP) minimizes a linear objective subject to multiple two-sided linear matrix inequalities intersected with a low-rank and spectral constrained domain set. Although solving LSOP is, in general, NP-hard, its partial convexification (i.e., replacing the domain set by its convex hull) termed “LSOP-R”, is often tractable and yields a high-quality solution. … Read more

On the Number of Pivots of Dantzig’s Simplex Methods for Linear and Convex Quadratic Programs

Refining and extending works by Ye and Kitahara-Mizuno, this paper presents new results on the number of pivots of simplex-type methods for solving linear programs of the Leontief kind, certain linear complementarity problems of the P kind, and nonnegative constrained convex quadratic programs. Our results contribute to the further understanding of the complexity and efficiency … Read more

On the longest chain of faces of completely positive and copositive cones

We consider a wide class of closed convex cones K in the space of real n*n symmetric matrices and establish the existence of a chain of faces of K, the length of which is maximized at n(n+1)/2 + 1. Examples of such cones include, but are not limited to, the completely positive and the copositive … Read more

Modeling risk for CVaR-based decisions in risk aggregation

Title Modeling risk for CVaR-based decisions in risk aggregation. Abstract Measuring the risk aggregation is an important exercise for any risk bearing carrier. It is not restricted to evaluation of the known portfolio risk position only, and could include complying with regulatory requirements, diversification, etc. The main difficulty of risk aggregation is creating an underlying … Read more

Worst-Case Conditional Value at Risk for Asset Liability Management: A Novel Framework for General Loss Functions

Asset-liability management (ALM) is a challenging task faced by pension funds due to the uncertain nature of future asset returns and interest rates. To address this challenge, this paper presents a new mathematical model that uses aWorst-case Conditional Value-at-Risk (WCVaR) constraint to ensure that the funding ratio remains above a regulator-mandated threshold with a high … Read more

Optimized Dimensionality Reduction for Moment-based Distributionally Robust Optimization

Moment-based distributionally robust optimization (DRO) provides an optimization framework to integrate statistical information with traditional optimization approaches. Under this framework, one assumes that the underlying joint distribution of random parameters runs in a distributional ambiguity set constructed by moment information and makes decisions against the worst-case distribution within the set. Although most moment-based DRO problems … Read more

The complexity of first-order optimization methods from a metric perspective

A central tool for understanding first-order optimization algorithms is the Kurdyka-Lojasiewicz inequality. Standard approaches to such methods rely crucially on this inequality to leverage sufficient decrease conditions involving gradients or subgradients. However, the KL property fundamentally concerns not subgradients but rather “slope”, a purely metric notion. By highlighting this view, and avoiding any use of … Read more

Water resources management: A bibliometric analysis and future research directions

The stochastic dual dynamic programming (SDDP) algorithm introduced by Pereira and Pinto in 1991 has sparked essential research in the context of water resources management, mainly due to its ability to address large-scale multistage stochastic problems. This paper aims to provide a tutorial-type review of 32 years of research since the publication of the SDDP … Read more