By using properties of dualizing parametrization functions, Lagrangian functions and the epigraph technique, some sufficient and necessary conditions of the stable strong duality and strong total duality for a class of DC optimization problems are established.

We discuss a partially augmented Lagrangian method for optimization programs with matrix inequality constraints. A global convergence result is obtained. Applications to hard problems in feedback control are presented to validate the method numerically.

In this paper we consider a block-structured convex optimization model, where in the objec-tive the block-variables are nonseparable and they are further linearly coupled in the constraint.For the 2-block case, we propose a number of first-order algorithms to solve this model. First,the alternating direction method of multipliers (ADMM) is extended, assuming that it is easyto op...

Distributed optimization is a fundamental mathematical theory for parallel and distributed systems. Several applications are normally designed based on such a theory, where parties cooperatively exchange messages with little or no central coordination to achieve some goals. In many situations, the transactions among the parties must be private, such as among members of social networks, hospital...

In this paper Mathematical structure of time-dependent Lagrangian systems and their symmetries are extended and the explicit relation between constants of motion and infinitesimal symmetries of time-dependent Lagrangian systems are considered. Starting point is time-independent Lagrangian systems ,then we extend mathematical concepts of these systems such as equivalent lagrangian systems to th...

During the past three decades, global optimization problems (including single-objective optimization problems (SOP) and multi-objective optimization problems (MOP)) have been intensively studied not only in Computer Science, but also in Engineering. There are many solutions in literature, such as gradient projection method [1-3], Lagrangian and augmented Lagrangian penalty methods [4-6], and ag...

In this paper, for nonconvex optimization problem with both equality and inequality constrains, we introduce a new augmented Lagrangian function and propose the corresponding multiplier algorithm. The global convergence is established without requiring the boundedness of multiplier sequences. In particular, if the algorithm terminates in finite steps, then we obtain a KKT point of the primal pr...

We propose a deterministic approach for global optimization of large-scale nonconvex quasiseparable problems encountered frequently in engineering systems design, such as multidisciplinary design optimization and product family optimization applications. Our branch and bound-based approach applies Lagrangian decomposition to 1) generate tight lower bounds by exploiting the structure of the prob...

The nuclear norm is widely used to induce low-rank solutions for many optimization problems with matrix variables. Recently, it has been shown that the augmented Lagrangian method (ALM) and the alternating direction method (ADM) are very efficient for many convex programming problems arising from various applications, provided that the resulting subproblems are sufficiently simple to have close...

Linearized alternating direction method of multipliers (ADMM) as an extension of ADMM has been widely used to solve linearly constrained problems in signal processing, machine leaning, communications, and many other fields. Despite its broad applications in non-convex optimization, for a great number of non-convex and non-smooth objective functions, its theoretical convergence guarantee is stil...