1,089 research outputs found
S-Lemma with Equality and Its Applications
Let and be two quadratic functions
having symmetric matrices and . The S-lemma with equality asks when the
unsolvability of the system implies the existence of a real
number such that . The
problem is much harder than the inequality version which asserts that, under
Slater condition, is unsolvable if and only if for some . In this paper, we
show that the S-lemma with equality does not hold only when the matrix has
exactly one negative eigenvalue and is a non-constant linear function
(). As an application, we can globally solve as well as the two-sided generalized trust region subproblem
without any condition. Moreover, the
convexity of the joint numerical range where is a (possibly non-convex) quadratic
function and are affine functions can be characterized
using the newly developed S-lemma with equality.Comment: 34 page
Tangential Extremal Principles for Finite and Infinite Systems of Sets, II: Applications to Semi-infinite and Multiobjective Optimization
This paper contains selected applications of the new tangential extremal
principles and related results developed in Part I to calculus rules for
infinite intersections of sets and optimality conditions for problems of
semi-infinite programming and multiobjective optimization with countable
constraint
Nonconvex Generalization of ADMM for Nonlinear Equality Constrained Problems
The ever-increasing demand for efficient and distributed optimization
algorithms for large-scale data has led to the growing popularity of the
Alternating Direction Method of Multipliers (ADMM). However, although the use
of ADMM to solve linear equality constrained problems is well understood, we
lacks a generic framework for solving problems with nonlinear equality
constraints, which are common in practical applications (e.g., spherical
constraints). To address this problem, we are proposing a new generic ADMM
framework for handling nonlinear equality constraints, neADMM. After
introducing the generalized problem formulation and the neADMM algorithm, the
convergence properties of neADMM are discussed, along with its sublinear
convergence rate , where is the number of iterations. Next, two
important applications of neADMM are considered and the paper concludes by
describing extensive experiments on several synthetic and real-world datasets
to demonstrate the convergence and effectiveness of neADMM compared to existing
state-of-the-art methods
Extended Formulations in Mixed-integer Convex Programming
We present a unifying framework for generating extended formulations for the
polyhedral outer approximations used in algorithms for mixed-integer convex
programming (MICP). Extended formulations lead to fewer iterations of outer
approximation algorithms and generally faster solution times. First, we observe
that all MICP instances from the MINLPLIB2 benchmark library are conic
representable with standard symmetric and nonsymmetric cones. Conic
reformulations are shown to be effective extended formulations themselves
because they encode separability structure. For mixed-integer
conic-representable problems, we provide the first outer approximation
algorithm with finite-time convergence guarantees, opening a path for the use
of conic solvers for continuous relaxations. We then connect the popular
modeling framework of disciplined convex programming (DCP) to the existence of
extended formulations independent of conic representability. We present
evidence that our approach can yield significant gains in practice, with the
solution of a number of open instances from the MINLPLIB2 benchmark library.Comment: To be presented at IPCO 201
- …