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We study a general convex optimization problem, which covers various classic problems in different areas and particularly includes many optimal transport related problems arising in recent years. To solve this problem, we revisit the classic Bregman proximal point algorithm (BPPA) and introduce a new inexact stopping condition for solving the subproblems, which can circumvent the underlying feasibility difficulty often appearing in existing inexact conditions when the problem has a complex feasible set. Our inexact condition also covers several existing inexact conditions and hence, as a byproduct, we actually develop a certain unified inexact framework for BPPA. This makes our inexact BPPA (iBPPA) more flexible to fit different scenarios in practice. In particular, as an application to the standard optimal transport (OT) problem, our iBPPA with the entropic proximal term can bypass some numerical instability issues that usually plague the well-recognized entropic regularization approach in the OT community, since our iBPPA does not require the proximal parameter to be very small for obtaining an accurate approximate solution. The iteration complexity of $mathcal{O}(1/k)$ and the convergence of the sequence are also established for our iBPPA under some mild conditions. Moreover, inspired by Nesterovs acceleration technique, we develop a variant of our iBPPA, denoted by V-iBPPA, and establish the iteration complexity of $mathcal{O}(1/k^{lambda})$, where $lambdageq1$ is a quadrangle scaling exponent of the kernel function. Some preliminary numerical experiments for solving the standard OT problem are conducted to show the convergence behaviors of our iBPPA and V-iBPPA under different inexactness settings. The experiments also empirically verify the potential of our V-iBPPA on improving the convergence speed.
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In this paper, we develop an inexact Bregman proximal gradient (iBPG) method based on a novel two-point inexact stopping condition, and establish the iteration complexity of $mathcal{O}(1/k)$ as well as the convergence of the sequence under some proper conditions. To improve the convergence speed, we further develop an inertial variant of our iBPG (denoted by v-iBPG) and show that it has the iteration complexity of $mathcal{O}(1/k^{gamma})$, where $gammageq1$ is a restricted relative smoothness exponent. Thus, when $gamma>1$, the v-iBPG readily improves the $mathcal{O}(1/k)$ convergence rate of the iBPG. In addition, for the case of using the squared Euclidean distance as the kernel function, we further develop a new inexact accelerated proximal gradient (iAPG) method, which can circumvent the underlying feasibility difficulty often appearing in existing inexact conditions and inherit all desirable convergence properties of the exact APG under proper summable-error conditions. Finally, we conduct some preliminary numerical experiments for solving a relaxation of the quadratic assignment problem to demonstrate the convergence behaviors of the iBPG, v-iBPG and iAPG under different inexactness settings.
We introduce a class of specially structured linear programming (LP) problems, which has favorable modeling capability for important application problems in different areas such as optimal transport, discrete tomography and economics. To solve these generally large-scale LP problems efficiently, we design an implementable inexact entropic proximal point algorithm (iEPPA) combined with an easy-to-implement dual block coordinate descent method as a subsolver. Unlike existing entropy-type proximal point algorithms, our iEPPA employs a more practically checkable stopping condition for solving the associated subproblems while achieving provable convergence. Moreover, when solving the capacity constrained multi-marginal optimal transport (CMOT) problem (a special case of our LP problem), our iEPPA is able to bypass the underlying numerical instability issues that often appear in the popular entropic regularization approach, since our algorithm does not require the proximal parameter to be very small in order to obtain an accurate approximate solution. Numerous numerical experiments show that our iEPPA is highly efficient and robust for solving large-scale CMOT problems, in comparison to the (stabilized) Dykstras algorithm and the commercial solver Gurobi. Moreover, the experiments on discrete tomography also highlight the potential modeling power of our model.
In this paper, an inexact proximal-point penalty method is studied for constrained optimization problems, where the objective function is non-convex, and the constraint functions can also be non-convex. The proposed method approximately solves a sequence of subproblems, each of which is formed by adding to the original objective function a proximal term and quadratic penalty terms associated to the constraint functions. Under a weak-convexity assumption, each subproblem is made strongly convex and can be solved effectively to a required accuracy by an optimal gradient-based method. The computational complexity of the proposed method is analyzed separately for the cases of convex constraint and non-convex constraint. For both cases, the complexity results are established in terms of the number of proximal gradient steps needed to find an $varepsilon$-stationary point. When the constraint functions are convex, we show a complexity result of $tilde O(varepsilon^{-5/2})$ to produce an $varepsilon$-stationary point under the Slaters condition. When the constraint functions are non-convex, the complexity becomes $tilde O(varepsilon^{-3})$ if a non-singularity condition holds on constraints and otherwise $tilde O(varepsilon^{-4})$ if a feasible initial solution is available.
In this paper, we consider an accelerated method for solving nonconvex and nonsmooth minimization problems. We propose a Bregman Proximal Gradient algorithm with extrapolation(BPGe). This algorithm extends and accelerates the Bregman Proximal Gradient algorithm (BPG), which circumvents the restrictive global Lipschitz gradient continuity assumption needed in Proximal Gradient algorithms (PG). The BPGe algorithm has higher generality than the recently introduced Proximal Gradient algorithm with extrapolation(PGe), and besides, due to the extrapolation step, BPGe converges faster than BPG algorithm. Analyzing the convergence, we prove that any limit point of the sequence generated by BPGe is a stationary point of the problem by choosing parameters properly. Besides, assuming Kurdyka-{L}ojasiewicz property, we prove the whole sequences generated by BPGe converges to a stationary point. Finally, to illustrate the potential of the new method BPGe, we apply it to two important practical problems that arise in many fundamental applications (and that not satisfy global Lipschitz gradient continuity assumption): Poisson linear inverse problems and quadratic inverse problems. In the tests the accelerated BPGe algorithm shows faster convergence results, giving an interesting new algorithm.
In this paper, we develop a parameterized proximal point algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent with a worst-case O(1/t) convergence rate, wheret denotes the iteration number. By properly choosing the algorithm parameters, numerical experiments on solving a sparse optimization problem arising from statistical learning show that our P-PPA could perform significantly better than other state-of-the-art methods, such as the alternating direction method of multipliers and the relaxed proximal point algorithm.
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