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Register a new userA Stochastic Sequential Quadratic Optimization Algorithm for Nonlinear Equality Constrained Optimization with Rank-Deficient Jacobians
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Added by
Frank E. Curtis
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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A sequential quadratic optimization algorithm is proposed for solving smooth nonlinear equality constrained optimization problems in which the objective function is defined by an expectation of a stochastic function. The algorithmic structure of the proposed method is based on a step decomposition strategy that is known in the literature to be widely effective in practice, wherein each search direction is computed as the sum of a normal step (toward linearized feasibility) and a tangential step (toward objective decrease in the null space of the constraint Jacobian). However, the proposed method is unique from others in the literature in that it both allows the use of stochastic objective gradient estimates and possesses convergence guarantees even in the setting in which the constraint Jacobians may be rank deficient. The results of numerical experiments demonstrate that the algorithm offers superior performance when compared to popular alternatives.
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We consider optimization problems on Riemannian manifolds with equality and inequality constraints, which we call Riemannian nonlinear optimization (RNLO) problems. Although they have numerous applications, the existing studies on them are limited especially in terms of algorithms. In this paper, we propose Riemannian sequential quadratic optimization (RSQO) that uses a line-search technique with an ell_1 penalty function as an extension of the standard SQO algorithm for constrained nonlinear optimization problems in Euclidean spaces to Riemannian manifolds. We prove its global convergence to a Karush-Kuhn-Tucker point of the RNLO problem by means of parallel transport and the exponential mapping. Furthermore, we establish its local quadratic convergence by analyzing the relationship between sequences generated by RSQO and the Riemannian Newton method. Ours is the first algorithm that has both global and local convergence properties for constrained nonlinear optimization on Riemannian manifolds. Empirical results show that RSQO finds solutions more stably and with higher accuracy compared with the existing Riemannian penalty and augmented Lagrangian methods.
Considering the constrained stochastic optimization problem over a time-varying random network, where the agents are to collectively minimize a sum of objective functions subject to a common constraint set, we investigate asymptotic properties of a distributed algorithm based on dual averaging of gradients. Different from most existing works on distributed dual averaging algorithms that mainly concentrating on their non-asymptotic properties, we not only prove almost sure convergence and the rate of almost sure convergence, but also asymptotic normality and asymptotic efficiency of the algorithm. Firstly, for general constrained convex optimization problem distributed over a random network, we prove that almost sure consensus can be archived and the estimates of agents converge to the same optimal point. For the case of linear constrained convex optimization, we show that the mirror map of the averaged dual sequence identifies the active constraints of the optimal solution with probability 1, which helps us to prove the almost sure convergence rate and then establish asymptotic normality of the algorithm. Furthermore, we also verify that the algorithm is asymptotically optimal. To the best of our knowledge, it seems to be the first asymptotic normality result for constrained distributed optimization algorithms. Finally, a numerical example is provided to justify the theoretical analysis.
Algorithm NCL is designed for general smooth optimization problems where first and second derivatives are available, including problems whose constraints may not be linearly independent at a solution (i.e., do not satisfy the LICQ). It is equivalent to the LANCELOT augmented Lagrangian method, reformulated as a short sequence of nonlinearly constrained subproblems that can be solved efficiently by IPOPT and KNITRO, with warm starts on each subproblem. We give numerical results from a Julia implementation of Algorithm NCL on tax policy models that do not satisfy the LICQ, and on nonlinear least-squares problems and general problems from the CUTEst test set.
This technical note proposes the decentralized-partial-consensus optimization with inequality constraints, and a continuous-time algorithm based on multiple interconnected recurrent neural networks (RNNs) is derived to solve the obtained optimization problems. First, the partial-consensus matrix originating from Laplacian matrix is constructed to tackle the partial-consensus constraints. In addition, using the non-smooth analysis and Lyapunov-based technique, the convergence property about the designed algorithm is further guaranteed. Finally, the effectiveness of the obtained results is shown while several examples are presented.
We propose a framework to use Nesterovs accelerated method for constrained convex optimization problems. Our approach consists of first reformulating the original problem as an unconstrained optimization problem using a continuously differentiable exact penalty function. This reformulation is based on replacing the Lagrange multipliers in the augmented Lagrangian of the original problem by Lagrange multiplier functions. The expressions of these Lagrange multiplier functions, which depend upon the gradients of the objective function and the constraints, can make the unconstrained penalty function non-convex in general even if the original problem is convex. We establish sufficient conditions on the objective function and the constraints of the original problem under which the unconstrained penalty function is convex. This enables us to use Nesterovs accelerated gradient method for unconstrained convex optimization and achieve a guaranteed rate of convergence which is better than the state-of-the-art first-order algorithms for constrained convex optimization. Simulations illustrate our results.
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