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Register a new userLinearly Convergent First-Order Algorithms for Semi-definite Programming
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In this paper, we consider two formulations for Linear Matrix Inequalities (LMIs) under Slater type constraint qualification assumption, namely, SDP smooth and non-smooth formulations. We also propose two first-order linearly convergent algorithms for solving these formulations. Moreover, we introduce a bundle-level method which converges linearly uniformly for both smooth and non-smooth problems and does not require any smoothness information. The convergence properties of these algorithms are also discussed. Finally, we consider a special case of LMIs, linear system of inequalities, and show that a linearly convergent algorithm can be obtained under a weaker assumption.
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Decentralized optimization is a powerful paradigm that finds applications in engineering and learning design. This work studies decentralized composite optimization problems with non-smooth regularization terms. Most existing gradient-based proximal decentralized methods are known to converge to the optimal solution with sublinear rates, and it remains unclear whether this family of methods can achieve global linear convergence. To tackle this problem, this work assumes the non-smooth regularization term is common across all networked agents, which is the case for many machine learning problems. Under this condition, we design a proximal gradient decentralized algorithm whose fixed point coincides with the desired minimizer. We then provide a concise proof that establishes its linear convergence. In the absence of the non-smooth term, our analysis technique covers the well known EXTRA algorithm and provides useful bounds on the convergence rate and step-size.
Many large-scale optimization problems can be expressed as composite optimization models. Accelerated first-order methods such as the fast iterative shrinkage-thresholding algorithm (FISTA) have proven effective for numerous large composite models. In this paper, we present a new variation of FISTA, to be called C-FISTA, which obtains global linear convergence for a broader class of composite models than many of the latest FISTA variants. We demonstrate the versatility and effectiveness of C-FISTA by showing C-FISTA outperforms current first-order solvers on both group Lasso and group logistic regression models. Furthermore, we utilize Fenchel duality to prove C-FISTA provides global linear convergence for a large class of convex models without the loss of global linear convergence.
In this paper, we propose two new solution schemes to solve the stochastic strongly monotone variational inequality problems: the stochastic extra-point solution scheme and the stochastic extra-momentum solution scheme. The first one is a general scheme based on updating the iterative sequence and an auxiliary extra-point sequence. In the case of deterministic VI model, this approach includes several state-of-the-art first-order methods as its special cases. The second scheme combines two momentum-based directions: the so-called heavy-ball direction and the optimism direction, where only one projection per iteration is required in its updating process. We show that, if the variance of the stochastic oracle is appropriately controlled, then both schemes can be made to achieve optimal iteration complexity of $mathcal{O}left(kappalnleft(frac{1}{epsilon}right)right)$ to reach an $epsilon$-solution for a strongly monotone VI problem with condition number $kappa$. We show that these methods can be readily incorporated in a zeroth-order approach to solve stochastic minimax saddle-point problems, where only noisy and biased samples of the objective can be obtained, with a total sample complexity of $mathcal{O}left(frac{kappa^2}{epsilon}lnleft(frac{1}{epsilon}right)right)$
In this paper, we aim to provide a comprehensive analysis on the linear rate convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex composite optimization problems. Under a certain error bound condition, we establish the global linear rate of convergence for a more general semi-proximal ADMM with the dual steplength being restricted to be in the open interval $(0, (1+sqrt{5})/2)$. In our analysis, we assume neither the strong convexity nor the strict complementarity except an error bound condition, which holds automatically for convex composite quadratic programming. This semi-proximal ADMM, which includes the classic ADMM, not only has the advantage to resolve the potentially non-solvability issue of the subproblems in the classic ADMM but also possesses the abilities of handling multi-block convex optimization problems efficiently. We shall use convex composite quadratic programming and quadratic semi-definite programming as important applications to demonstrate the significance of the obtained results. Of its own novelty in second-order variational analysis, a complete characterization is provided on the isolated calmness for the nonlinear convex semi-definite optimization problem in terms of its second order sufficient optimality condition and the strict Robinson constraint qualification for the purpose of proving the linear rate convergence of the semi-proximal ADMM when applied to two- and multi-block convex quadratic semi-definite programming.
In this paper we consider a class of convex conic programming. In particular, we propose an inexact augmented Lagrangian (I-AL) method for solving this problem, in which the augmented Lagrangian subproblems are solved approximately by a variant of Nesterovs optimal first-order method. We show that the total number of first-order iterations of the proposed I-AL method for computing an $epsilon$-KKT solution is at most $mathcal{O}(epsilon^{-7/4})$. We also propose a modified I-AL method and show that it has an improved iteration-complexity $mathcal{O}(epsilon^{-1}logepsilon^{-1})$, which is so far the lowest complexity bound among all first-order I-AL type of methods for computing an $epsilon$-KKT solution. Our complexity analysis of the I-AL methods is mainly based on an analysis on inexact proximal point algorithm (PPA) and the link between the I-AL methods and inexact PPA. It is substantially different from the existing complexity analyses of the first-order I-AL methods in the literature, which typically regard the I-AL methods as an inexact dual gradient method. Compared to the mostly related I-AL methods cite{Lan16}, our modified I-AL method is more practically efficient and also applicable to a broader class of problems.
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