Double-Regularization Proximal Methods, with Complementarity Applications

We consider the variational inequality problem formed by a general set-valued maximal monotone operator and a possibly unbounded “box” in , and study its solution by proximal methods whose distance regularizations are coercive over the box. We prove convergence for a class of double regularizations generalizing a previously-proposed class of Auslender et al. Using these results, we derive a broadened class of augmented Lagrangian methods. We point out some connections between these methods and earlier work on “pure penalty” smoothing methods for complementarity; this connection leads to a new form of augmented Lagrangian based on the “neural” smoothing function. Finally, we computationally compare this new kind of augmented Lagrangian to three previously-known varieties on the MCPLIB problem library, and show that the neural approach offers some advantages. In these tests, we also consider primal-dual approaches that include a primal proximal term. Such a stabilizing term tends to slow down the algorithms, but makes them more robust. This author was partially supported by CNPq, Grant PQ 304133/2004-3 and PRONEX-Optimization.     

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

Parametric convex programming has received a lot of attention, since it has many applications in chemical engineering, control engineering, signal processing, etc. Further, inverse optimality plays an important role in many contexts, e.g., image processing, motion planning. This paper introduces a constructive solution of the inverse optimality problem for the class of continuous piecewise affine functions. The main idea is based on the convex lifting concept. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: Any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Regarding linear optimal control, it will be shown that any continuous piecewise affine control law can be obtained via a linear optimal control problem with the control horizon at most equal to 2 prediction steps.     

Fundamental Limits of Weak Recovery with Applications to Phase Retrieval

Foundations of Computational Mathematics - In phase retrieval, we want to recover an unknown signal $${{\varvec{x}}}\in {{\mathbb {C}}}^d$$ from n quadratic measurements of the form $$y_i =...     

Carleman Estimates and Applications to Inverse Problems

No Abstract. .Lecture held in the Seminario Matematico e Fisico on March 18, 2004Received: May, 2004     

Sequential Quadratic Programming Methods for Large-Scale Problems

Sequential quadratic (SQP) programming methodsare the method of choice when solving small or medium-sized problems. Sincethey are complex methods they are difficult (but not impossible) to adapt tosolve large-scale problems. We start by discussing the difficulties that needto be addressed and then describe some general ideas that may be used toresolve these difficulties. A number of SQP codes have been written to solve specific applications and there is a general purposed SQP code called SNOPT,which is intended for general applications of a particular type. These aredescribed briefly together with the ideas on which they are based. Finally wediscuss new work on developing SQP methods using explicit second derivatives.     

凸二次规划问题逆问题的模型与解法

本文分别考虑带非负约束和不带大量负约束凸二次规划问题逆问题。首先得到各个逆问题的数学模型,然后对不同的模型给出不同的求解方法。     

Inertial Proximal ADMM for Separable Multi-Block Convex Optimizations and Compressive Affine Phase Retrieval

Acta Mathematica Sinica, English Series - Separable multi-block convex optimization problem appears in many mathematical and engineering fields. In the first part of this paper, we propose an...     

随机系数下线性二次控制问题的最优性条件及其应用*

本文旨在研究随机系数下随机微分方程的线性二次最优控制问题.本文从闭环最优控制/策略存在的必要性条件的角度开展研究. 若闭环最优控制/策略存在, 得到其显示反馈表示、带伪逆运算的倒向随机Riccati方程的适定性及不同系数间满足的一些本质性条件. 此处结论本质地推广和改进了文[Ait Rami M, Moore J, Zhou X. Indefinite stochastic linear quadratic control and generalized differential Riccati equation [J]. {\it SIAM J Control Optim,} 2001, 40:1296--1311;Sun J, Yong J. Linear quadratic stochastic differential games: open-loop and closed-loop saddle points [J]. {\it SIAM J Control Optim,} 2014, 52:4082--4121;L\"{u} Q, Wang T, Zhang X. Characterization of optimal feedback for stochastic linear quadratic control problems,Probab Uncertain Quant Risk, 2017, 2017, 2:11, DOI 10.1186/s41546-017-0022-7]的相应结论.此外, 本文得到了一个关于倒向随机Riccati方程和二阶伴随方程两类方程适应解之间的微妙关系. 注意到,这一结论在现有文献中首次出现. 最后, 本文讨论了在均值方差对冲问题中的应用.     

Global Optimality Conditions and Optimization Methods for Quadratic Knapsack Problems

The quadratic knapsack problem (QKP) maximizes a quadratic objective function subject to a binary and linear capacity constraint. Due to its simple structure and challenging difficulty, it has been studied intensively during the last two decades. This paper first presents some global optimality conditions for (QKP), which include necessary conditions and sufficient conditions. Then a local optimization method for (QKP) is developed using the necessary global optimality condition. Finally a global optimization method for (QKP) is proposed based on the sufficient global optimality condition, the local optimization method and an auxiliary function. Several numerical examples are given to illustrate the efficiency of the presented optimization methods.     

Inexact Proximal Point Methods for Equilibrium Problems in Banach Spaces

We introduce two inexact proximal-like methods for solving equilibrium problems in reflexive Banach spaces and establish their convergence properties, proving that the sequence generated by each one of them converges to a solution of the equilibrium problem under reasonable assumptions.     

Optimization Methods for Solving Inverse Immunology and Epidemiology Problems

Computational Mathematics and Mathematical Physics - Inverse problems for systems of nonlinear ordinary differential equations are studied. In these problems, the unknown coefficients and...     

On Regularization Methods for Inverse Problems of Dynamic Type

In this paper, we consider new regularization methods for linear inverse problems of dynamic type. These methods are based on dynamic programming techniques for linear quadratic optimal control problems. Two different approaches are followed: a continuous and a discrete one. We prove regularization properties and also obtain rates of convergence for the methods derived from both approaches. A numerical example concerning the dynamic EIT problem is used to illustrate the theoretical results.     

Hybrid Proximal Point Algorithm and Applications to Equilibrium Problems and Convex Programming

The problem of finding minima of weakly sequentially lower semicontinuous functions on reflexive Banach spaces is studied by means of convex and nonconvex subdifferentials. Finding a descent direction for a non-stationary point is a question of importance for many optimization algorithms. The existence or non-existence of such a direction is clarified through several theorems and a series of selective examples. For the general problem, a notion called radius of descent is proposed and shown to be useful for the analysis related to descent directions.     

均衡问题和不动点问题的粘滞近似方法及其应用

用粘滞近似方法产生了一个新的迭代序列,并证明了该迭代序列强收敛于一个非扩张映射的不动点,同时该不动点也是一个变分不等式和一个均衡问题的共同解.作为应用,另外证明了一个关于非扩张映射和严格伪压缩映射的定理.     

Quadratic Majorization for Nonconvex Loss with Applications to the Boosting Algorithm

Classical robust statistical methods dealing with noisy data are often based on modifications of convex loss functions. In recent years, nonconvex loss-based robust methods have been increasingly popular. A nonconvex loss can provide robust estimation for data contaminated with outliers. The significant challenge is that a nonconvex loss can be numerically difficult to optimize. This article proposes quadratic majorization algorithm for nonconvex (QManc) loss. The QManc can decompose a nonconvex loss into a sequence of simpler optimization problems. Subsequently, the QManc is applied to a powerful machine learning algorithm: quadratic majorization boosting algorithm (QMBA). We develop QMBA for robust classification (binary and multi-category) and regression. In high-dimensional cancer genetics data and simulations, the QMBA is comparable with convex loss-based boosting algorithms for clean data, and outperforms the latter for data contaminated with outliers. The QMBA is also superior to boosting when directly implemented to optimize nonconvex loss functions. Supplementary material for this article is available online.     

Parallel Implementation of Successive Convex Relaxation Methods for Quadratic Optimization Problems

As computing resources continue to improve, global solutions for larger size quadrically constrained optimization problems become more achievable. In this paper, we focus on larger size problems and get accurate bounds for optimal values of such problems with the successive use of SDP relaxations on a parallel computing system called Ninf (Network-based Information Library for high performance computing).     

Approximation Methods for Inverse Problems Governed by Nonlinear Parabolic Systems

We present a rigorous theoretical framework for approximation of nonlinear parabolic systems with delays in the context of inverse least squares problems. Convergence of approximate optimal parameters and that of forward solutions in the context of semidiscrete Galerkin schemes are given. Sample numerical results demonstrating the convergence are given for a model of dioxin uptake and elimination in a distributed liver model that is a special case of the general theoretical framework.