2

Sampling-based methods for multi-block optimization problems over transport polytopes
This paper focuses on multi-block optimization problems over transport polytopes, which underlie various applications including …
H., M. Li, X. Liu, C. Meng
Cite DOI Preprint
The exactness of the $\ell_1$ penalty function for a class of mathematical programs with generalized complementarity constraints

We establish the exactness of $l_1$ penalty function for a class of mathematical programs with generalized complementarity constraints (MPGCC). MPGCC extends the notion of mathematical programs with complementarity constraints (MPCC), in that every two block variables complement each other.

MP(G)CC is notorious for its violation of commonly used constraint qualifications (e.g., Mangasarian-Fromovitz constraint qualification), rendering the Karush-Kuhn-Tucker conditions no longer necessary. A simple treatment for this issue is to penalize the complementarity constraints in $l_1$ form. It is thus of importance to study the exactness of the penalty function, i.e., whether the penalty problem shares the same optimal solution set with the original one given a sufficiently large penalty parameter.

Setting

We focus our attention on a class of MPGCCs with multi-affine objective functions and separable polytopic feasible sets. The problem class applies to the network transportation and the understanding of strongly correlated quantum systems (see our paper for details). Existing works either impose stringent assumptions or fail to cover the nonlinear and multi-block setting.

Results

Without extra assumptions, we establish the exactness by fully exploiting the problem structure.

H., X. Liu
Cite DOI Preprint
Projected gradient descent algorithm for $\textit{ab initio}$ crystal structure relaxation under a fixed unit cell volume
This paper is concerned with ab initio crystal structure relaxation under a fixed unit cell volume, which is a step in calculating the …
H., J. Yin, X. Gao, X. Liu, H. Song
Cite DOI Preprint
The convergence properties of infeasible inexact proximal alternating linearized minimization

We study the theoretical properties of proximal alternating linearized minimization (PALM) method, which is a popular unified framework for multi-block nonconvex optimization.

Setting

The subproblems are solved by infeasible subsolvers, yielding infeasible iterates. Examples are ubiquitous; for example, consider a feasible set described by a severely under-determined linear system. However, there are few existing works in this aspect.

Theoretical results
  • Implementable inexact criterion for solving subproblems based on error bounds.
  • Global convergence to stationarity and novel asymptotic convergence rates.
H., X. Liu
Cite DOI Preprint
A global optimization approach for multimarginal optimal transport problems with Coulomb cost

We develop an optimization approach for the multimarginal optimal transport problems with Coulomb cost (MMOT), arising as the strong-interaction limit of density functional theory (DFT). This work may contribute to a better understanding of strongly correlated electrons systems, for which the conventional Kohn-Sham DFT induces large systematic errors.

Setting

We consider the MMOT under a Monge-like ansatz. The Monge-like ansatz explicitly describes the coupling between electron positions, amounts to a spectacular dimension reduction, and generalizes the famous Monge ansatz in the OT community. When the number of electrons exceeds 2, the MMOT turns into a nonconvex optimization problem. As such, existing works employing the Monge-like ansatz limit the focus to two-electron systems.

Method

We develop a global optimization approach for the nonconvex problem, featuring
Results
  • Simulation of typical systems with up to 7 electrons and over 10x the existing discretization scale. The results conform to theoretical predictions and physical intuitions.
  • First visualization of mappings between electron positions in two-dimensional context.
H., H. Chen, X. Liu
Cite DOI Preprint
Force-based gradient descent method for $\textit{ab initio}$ atomic structure relaxation

We develop a force-based nonmonotone method for atomic structure relaxation, a ubiquitous yet expensive task in materials simulation and design. Conventional methods such as conjugate gradient methods are computationally robust but usually require a lengthy relaxation process, resulting in their inefficiency.

Method

The inefficiency of conjugate gradient methods can be credited to the improper line minimization (LM) process therein. We devise a novel method named WANBB with an improved LM process, featuring

  • curvature-informed initial trial step sizes;
  • a reweighted nonmonotone line minimization criterion.

The convergence of WANBB to equilibrium is also rigorously established.

Results

We test WANBB against conjugate gradient methods and quasi-Newton methods over a benchmark set containing nearly 80 systems with good universality.

  • Convergence of WANBB on all systems (with a high-entropy alloy as demonstration).
  • Speedup factor over conjugate gradient methods > 1.5.

Important Note

WANBB has been integrated into a suite for crystal structure relaxation called ProMe-SuRe, which can be interfaced to some widely used materials simulation software. Static-link library files are available upon request!
H., X. Gao, Y. Zhao, X. Liu, H. Song
Cite DOI Preprint