In this paper we investigate how standard nonlinear programming algorithms can be used to solve constrained optimization problems in a distributed manner. The optimization setup consists of a set of agents interacting through a communication graph that have as common goal the minimization of a function expressed as a sum of (possibly non-convex) differentiable functions. Each function in the sum corresponds to an agent and each agent has associated an equality constraint. By re-casting the distributed optimization problem into an equivalent, augmented centralized problem, we show that distributed algorithms result naturally from applying standard nonlinear programming techniques. Due to the distributed formulation, the standard assumptions and convergence results no longer hold. We emphasize what changes are necessary for convergence to still be achieved for three algorithms: two algorithms based on Lagrangian methods, and an algorithm based the method of multipliers. The changes in the convergence results are necessary mainly due to the fact that the (local) minimizers of the lifted optimization problem are not regular, as a results of the distributed formulation. Unlike the standard algorithm based on the method of multipliers, for the distributed version we cannot show that the theoretical superlinear convergence rate can be achieved.
This paper develops distributed synchronous and asynchronous algorithms for the large-scale semi-definite programming with diagonal constraints, which has wide applications in combination optimization, image processing and community detection. The information of the semi-definite programming is allocated to multiple interconnected agents such that each agent aims to find a solution by communicating to its neighbors. Based on low-rank property of solutions and the Burer-Monteiro factorization, we transform the original problem into a distributed optimization problem over unit spheres to reduce …
This paper presents the Distributed Primal Outer Approximation (DiPOA) algorithm for solving Sparse Convex Programming (SCP) problems with separable structures, efficiently, and in a decentralized manner. The DiPOA algorithm development consists of embedding the recently proposed Relaxed Hybrid Alternating Direction Method of Multipliers (RH-ADMM) algorithm into the Outer Approximation (OA) algorithm. We also propose two main improvements to control the quality and the number of cutting planes that approximate nonlinear functions. In particular, the RH-ADMM algorithm acts as a distributed nume…
Superlinear and quadratic convergence of a stabilized sequential quadratic semidefinite programming method for nonlinear semidefinite programming problems
We present new constraint qualification conditions for nonlinear semidefinite programming that extend some of the constant rank-type conditions from nonlinear programming. As an application of these conditions, we provide a unified global convergence proof of a class of algorithms to stationary points without assuming neither uniqueness of the Lagrange multiplier nor boundedness of the Lagrange multipliers set. This class of algorithm includes, for instance, general forms of augmented Lagrangian, sequential quadratic programming, and interior point methods. We also compare these new conditions…
In this paper, we present a two phase method for solving nonlinear programming problems called Nonlinear Polyhedral Active Set Algorithm (NPASA) that has global and local convergence guarantees under reasonable assumptions. The first phase consists of an augmented Lagrangian method to ensure global convergence while the second phase is designed to promote fast local convergence by performing a balanced reduction of two error estimators for nonlinear programs. After presenting error estimators for nonlinear programs and our algorithm NPASA, we establish global convergence properties for NPASA. …
In this paper, we extend the class of strongly $E$-preinvex and strongly $E$-invex functions to quasi strongly $E$-preinvex, quasi strongly $E$-invex and pseudo strongly $E$-invex functions. Some nontrivial suitable examples have been constructed in support of our definitions. Several interesting properties and relationships of these functions are discussed. Furthermore, to show the application of our results, we consider a nonlinear programming problem and show that the local minimum point is also a strictly global minimum.
In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under $\tilde{G}$-expectation. Under standard assumptions, we establish the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution of a type of fully nonlinear HJB equation.
In this paper, we accomplish a unified convergence analysis of a second-order method of multipliers (i.e., a second-order augmented Lagrangian method) for solving the conventional nonlinear conic optimization problems.Specifically, the algorithm that we investigated incorporates a specially designed nonsmooth (generalized) Newton step to furnish a second-order update rule for the multipliers.We first show in a unified fashion that under a few abstract assumptions, the proposed method is locally convergent and possesses a (nonasymptotic) superlinear convergence rate, even though the penalty par…
In this paper, we provide local convergence analysis for the two phase Nonlinear Polyhedral Active Set Algorithm (NPASA) designed to solve nonlinear programs. In particular, we establish local quadratic convergence of the primal iterates and global error estimator for NPASA under reasonable assumptions. Additionally, under the same set of assumptions we prove that only phase two of NPASA is executed after finitely many iterations. This paper is companion to a paper that provides motivation and global convergence analysis for NPASA.
Second-order necessary optimality conditions for nonlinear conic programming problems that depend on a single Lagrange multiplier are usually built under nondegeneracy and strict complementarity. In this paper we establish a condition of such type for two classes of nonlinear conic problems, namely semidefinite and second-order cone programming, assuming Robinson's constraint qualification and a weak constant rank-type property which are, together, strictly weaker than nondegeneracy. Our approach is done via a penalty-based strategy, which is aimed at providing strong global convergence result…
The augmented Lagrangian method (ALM) has gained tremendous popularity for its elegant theory and impressive numerical performance since it was proposed by Hestenes and Powell in 1969. It has been widely used in numerous efficient solvers to improve numerical performance to solve many problems. In this paper, without requiring the uniqueness of multipliers, the local (asymptotic Q-superlinear) Q-linear convergence rate of the primal-dual sequences generated by ALM for the nonlinear semidefinite programming (NLSDP) is established by assuming the second-order sufficient condition (SOSC) and the …
In this paper, we propose two exact distributed algorithms to solve mixed integer linear programming (MILP) problems with multiple agents where data privacy is important for the agents. A key challenge is that, because of the non-convex nature of MILPs, classical distributed and decentralized optimization approaches cannot be applied directly to find their optimal solutions. The proposed exact algorithms are based on adding primal cuts and restricting the Lagrangian relaxation of the original MILP problem. We show finite convergence of these algorithms for MILPs with only binary and continuous…
This paper proposes a new state estimator for discrete-time nonlinear dynamical systems with unknown-but-bounded uncertainties and state linear inequality and nonlinear equality constraints. Our algorithm is based on constrained zonotopes (CZs) and on a DC programming approach (DC stands for difference of convex functions). Recently, mean value extension and first-order Taylor extension have been adapted from zonotopes to propagate CZs over nonlinear mappings. Although the resulting algorithms (called CZMV and CZFO) reach better precision than the original zonotopic versions, they carry the se…
In this paper, we propose a new sequential quadratic semidefinite programming (SQSDP) method for solving degenerate nonlinear semidefinite programs (NSDPs), in which we produce iteration points by solving a sequence of stabilized quadratic semidefinite programming (QSDP) subproblems, which we derive from the minimax problem associated with the NSDP. Unlike the existing SQSDP methods, the proposed one allows us to solve those QSDP subproblems inexactly, and each QSDP is feasible. One more remarkable point of the proposed method is that constraint qualifications (CQs) or boundedness of Lagrange …
The Piecewise Polynomial Interpolation (PPI) function approach is aimed at solving nonlinear programming problems with disjoint feasible regions. In such problems, disjointedness is generally associated with prohibited operating zones, which correspond to bands of values that a variable is not allowed to assume. An analytical implication of such prohibited operating zones is to make the objective function, as well as its domain, discontinuous. The PPI function approach consists in replacing the constraints associated with prohibited operating zones by an equivalent set of equality and inequali…
In [R. Andreani, G. Haeser, L. M. Mito, H. Ramírez C., Weak notions of nondegeneracy in nonlinear semidefinite programming, arXiv:2012.14810, 2020] the classical notion of nondegeneracy (or transversality) and Robinson's constraint qualification have been revisited in the context of nonlinear semidefinite programming exploiting the structure of the problem, namely, its eigendecomposition. This allows formulating the conditions equivalently in terms of (positive) linear independence of significantly smaller sets of vectors. In this paper we extend these ideas to the context of nonlinear second-…
In 2020, Yamakawa and Okuno proposed a stabilized sequential quadratic semidefinite programming (SQSDP) method for solving, in particular, degenerate nonlinear semidefinite optimization problems. The algorithm is shown to converge globally without a constraint qualification, and it has some nice properties, including the feasible subproblems, and their possible inexact computations. In particular, the convergence was established for approximate-Karush-Kuhn-Tucker (AKKT) and trace-AKKT conditions, which are two sequential optimality conditions for the nonlinear conic contexts. However, recently…
