Devising efficient algorithms to solve continuously-varying strongly convex optimization programs is key in many applications, from control systems to signal processing and machine learning. In this context, solving means to find and track the optimizer trajectory of the continuously-varying convex optimization program. Recently, a novel prediction-correction methodology has been put forward to set up iterative algorithms that sample the continuously-varying optimization program at discrete time steps and perform a limited amount of computations to correct their approximate optimizer with the new sampled problem and predict how the optimizer will change at the next time step. Prediction-correction algorithms have been shown to outperform more classical strategies, i.e., correction-only methods. Typically, prediction-correction methods have asymptotic tracking errors of the order of $h^2$, where $h$ is the sampling period, whereas classical strategies have order of $h$. Up to now, Prediction-correction algorithms have been developed in the primal space, both for unconstrained and simply constrained convex programs. In this paper, we show how to tackle linearly constrained continuously-varying problem by prediction-correction in the dual space and we prove similar asymptotic error bounds as their primal versions.
We present a method for solving linearly constrained convex optimization problems, which is based on the application of known algorithms for finding zeros of the sum of two monotone operators (presented by Eckstein and Svaiter) to the dual problem. We establish convergence rates for the new method, and we present applications to TV denoising and compressed sensing problems.
Motivated by TRACE algorithm [Curtis et al. 2017], we propose a trust region algorithm for finding second order stationary points of a linearly constrained non-convex optimization problem. We show the convergence of the proposed algorithm to (ε_g, ε_H)-second order stationary points in \widetilde{\mathcal{O}}(\max{ε_g^{-3/2}, ε_H^{-3}}) iterations. This iteration complexity is achieved for general linearly constrained optimization without cubic regularization of the objective function.
We study the \emph{Proximal Alternating Predictor-Corrector} (PAPC) algorithm introduced recently by Drori, Sabach and Teboulle to solve nonsmooth structured convex-concave saddle point problems consisting of the sum of a smooth convex function, a finite collection of nonsmooth convex functions and bilinear terms. We introduce the notion of pointwise quadratic supportability, which is a relaxation of a standard strong convexity assumption and allows us to show that the primal sequence is R-linearly convergent to an optimal solution and the primal-dual sequence is globally Q-linearly convergent…
In this paper, we establish the convergence properties for a majorized alternating direction method of multipliers (ADMM) for linearly constrained convex optimization problems whose objectives contain coupled functions. Our convergence analysis relies on the generalized Mean-Value Theorem which plays an important role to properly control the cross terms due to the presence of coupled objective functions. Our results in particular show that directly applying 2-block ADMM with a large step length to the linearly constrained convex optimization problem with a quadratically coupled objective funct…
In a real Hilbert space setting, we reconsider the classical Arrow-Hurwicz differential system in view of solving linearly constrained convex minimization problems. We investigate the asymptotic properties of the differential system and provide conditions for which its solutions converge towards a saddle point of the Lagrangian associated with the convex minimization problem. Our convergence analysis mainly relies on a `Lagrangian identity' which naturally extends on the well-known descent property of the classical continuous steepest descent method. In addition, we present asymptotic estimate…
This paper presents a majorized alternating direction method of multipliers (ADMM) with indefinite proximal terms for solving linearly constrained $2$-block convex composite optimization problems with each block in the objective being the sum of a non-smooth convex function and a smooth convex function, i.e., $\min_{x \in {\cal X}, \; y \in {\cal Y}}\{p(x)+f(x) + q(y)+g(y)\mid A^* x+B^* y = c\}$. By choosing the indefinite proximal terms properly, we establish the global convergence and $O(1/k)$ ergodic iteration-complexity of the proposed method for the step-length $τ\in (0, (1+\sqrt{5})/2)$.…
This work proposes an accelerated primal-dual dynamical system for affine constrained convex optimization and presents a class of primal-dual methods with nonergodic convergence rates. In continuous level, exponential decay of a novel Lyapunov function is established and in discrete level, implicit, semi-implicit and explicit numerical discretizations for the continuous model are considered sequentially and lead to new accelerated primal-dual methods for solving linearly constrained optimization problems. Special structures of the subproblems in those schemes are utilized to develop efficient …
Most recently, He and Yuan [arXiv:2108.08554, 2021] have proposed a balanced augmented Lagrangian method (ALM) for the canonical convex programming problem with linear constraints, which advances the original ALM by balancing its subproblems and improving its implementation. In this short note, we propose a dual-primal version of the balanced ALM, which updates the new iterate via a conversely dual-primal iterative order formally. The proposed method inherits all advantages of the prototype balanced ALM, and its convergence analysis can be well conducted in the context of variational inequalit…
In this paper, we consider large-scale linearly constrained composite convex optimization problem, whose objective is a sum of a smooth function and a possibly nonsmooth function. We propose a scalable \textbf{F}rank-\textbf{W}olfe based \textbf{A}ugmented \textbf{L}agrangian (FW-AL) method for solving this problem. At each iteration, the proposed FW-AL method employs the FW method (or its variants) to approximately solve the AL subproblem {(with fixed Lagrange multiplier)} within a preselected tolerance and then updates the Lagrange multiplier. The proposed FW-AL method is well suitable for s…
This paper establishes the iteration-complexity of a Jacobi-type non-Euclidean proximal alternating direction method of multipliers (ADMM) for solving multi-block linearly constrained nonconvex programs. The subproblems of this ADMM variant can be solved in parallel and hence the method has great potential to solve large scale multi-block linearly constrained nonconvex programs. Moreover, our analysis allows the Lagrange multiplier to be updated with a relaxation parameter in the interval (0, 2).
This paper aims to study a majorized alternating direction method of multipliers with indefinite proximal terms (iPADMM) for convex composite optimization problems. We show that the majorized iPADMM for 2-block convex optimization problems converges globally under weaker conditions than those used in the literature and exhibits a linear convergence rate under a local error bound condition. Based on these, we establish the linear rate convergence results for a symmetric Gaussian-Seidel based majorized iPADMM, which is designed for multi-block composite convex optimization problems. Moreover, we…
In this paper we combine the stochastic variance reduced gradient (SVRG) method [17] with the primal dual fixed point method (PDFP) proposed in [7] to solve a sum of two convex functions and one of which is linearly composite. This type of problems are typically arisen in sparse signal and image reconstruction. The proposed SVRG-PDFP can be seen as a generalization of Prox-SVRG [37] originally designed for the minimization of a sum of two convex functions. Based on some standard assumptions, we propose two variants, one is for strongly convex objective function and the other is for general con…
In this paper we propose a distributed dual gradient algorithm for minimizing linearly constrained separable convex problems and analyze its rate of convergence. In particular, we prove that under the assumption of strong convexity and Lipshitz continuity of the gradient of the primal objective function we have a global error bound type property for the dual problem. Using this error bound property we devise a fully distributed dual gradient scheme, i.e. a gradient scheme based on a weighted step size, for which we derive global linear rate of convergence for both dual and primal suboptimality…
By time discretization of a second-order primal-dual dynamical system with damping $α/t$ where an inertial construction in the sense of Nesterov is needed only for the primal variable, we propose a fast primal-dual algorithm for a linear equality constrained convex optimization problem. Under a suitable scaling condition, we show that the proposed algorithm enjoys a fast convergence rate for the objective residual and the feasibility violation, and the decay rate can reach $\mathcal{O}(1/k^{α-1})$ at the most. We also study convergence properties of the corresponding primal-dual dynamical syst…
This note serves two purposes. Firstly, we construct a counterexample to show that the statement on the convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex optimization problems in a highly influential paper by Boyd et al. [Found. Trends Mach. Learn. 3(1) 1-122 (2011)] can be false if no prior condition on the existence of solutions to all the subproblems involved is assumed to hold. Secondly, we present fairly mild conditions to guarantee the existence of solutions to all the subproblems and provide a rigorous convergence analysis on t…
The generalized alternating direction method of multipliers (ADMM) of Xiao et al. [{\tt Math. Prog. Comput., 2018}] aims at the two-block linearly constrained composite convex programming problem, in which each block is in the form of "nonsmooth + quadratic". However, in the case of non-quadratic (but smooth), this method may fail unless the favorable structure of 'nonsmooth + smooth' is no longer used. This paper aims to remedy this defect by using a majorized technique to approximate the augmented Lagrangian function, so that the corresponding subprobllem can be decomposed into some smaller …
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…
We propose an inexact proximal augmented Lagrangian framework with explicit inner problem termination rule for composite convex optimization problems. We consider arbitrary linearly convergent inner solver including in particular stochastic algorithms, making the resulting framework more scalable facing the ever-increasing problem dimension. Each subproblem is solved inexactly with an explicit and self-adaptive stopping criterion, without requiring to set an a priori target accuracy. When the primal and dual domain are bounded, our method achieves $O(1/\sqrtε)$ and $O(1/ε)$ complexity bound in…
In this paper we consider a distributed convex optimization problem over time-varying networks. We propose a dual method that converges R-linearly to the optimal point given that the agents' objective functions are strongly convex and have Lipschitz continuous gradients. The proposed method requires half the amount of variable exchanges per iterate than methods based on DIGing, and yields improved practical performance as empirically demonstrated.
This document introduces a strategy to solve linear optimization problems. The strategy is based on the bounding condition each constraint produces on each one of the problem's dimension. The solution of a linear optimization problem is located at the intersection of the constraints defining the extreme vertex. By identifying the constraints that limit the growth of the objective function value, we formulate linear equations system leading to the optimization problem's solution.The most complex operation of the algorithm is the inversion of a matrix sized by the number of dimensions of the pro…
Consider the problem of minimizing the sum of a smooth convex function and a separable nonsmooth convex function subject to linear coupling constraints. Problems of this form arise in many contemporary applications including signal processing, wireless networking and smart grid provisioning. Motivated by the huge size of these applications, we propose a new class of first order primal-dual algorithms called the block successive upper-bound minimization method of multipliers (BSUM-M) to solve this family of problems. The BSUM-M updates the primal variable blocks successively by minimizing local…
This paper analyzes the iteration-complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs. More specifically, the objective function is of the form $f + h$ where $f$ is a differentiable function whose gradient is Lipschitz continuous and $h$ is a closed convex function with bounded domain. The method, basically, consists of applying an accelerated inexact proximal point method for solving approximately a sequence of quadratic penalized subproblems associated to the linearly constrained problem. Each subproblem of…
We consider the problem of minimizing a function, which is the sum of a linear function and a composition of a strongly convex function with a linear transformation, over a compact polyhedral set. Jaggi and Lacoste-Julien [14] showed that the conditional gradient method with away steps employed on the aforementioned problem without the additional linear term has linear rate of convergence, depending on the so-called pyramidal width of the feasible set. We revisit this result and provide a variant of the algorithm and an analysis that is based on simple duality arguments, as well as correspondi…
This paper studies robust solutions and semidefinite linear programming (SDP) relaxations of a class of convex polynomial programs in the face of data uncertainty. The class of convex programs, called robust SOS-convex programs, includes robust quadratically constrained convex programs and robust separable convex polynomial programs. It establishes sums of squares polynomial representations characterizing robust solutions and exact SDP-relaxations of robust SOS-convex programs under various commonly used uncertainty sets. In particular, the results show that the polytopic and ellipsoidal uncer…
In this paper, we present difference of convex algorithms for solving bilevel programs in which the upper level objective functions are difference of convex functions, and the lower level programs are fully convex. This nontrivial class of bilevel programs provides a powerful modelling framework for dealing with applications arising from hyperparameter selection in machine learning. Thanks to the full convexity of the lower level program, the value function of the lower level program turns out to be convex and hence the bilevel program can be reformulated as a difference of convex bilevel prog…
In this paper we introduce a new dual program, which is representable as a semi-definite linear programming problem, for a primal convex minimax programming model problem and show that there is no duality gap between the primal and the dual whenever the functions involved are SOS-convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust SOS-convex programming problems under data uncertainty and to minimax fractional programming problems with SOS-convex polyno…
A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy point computation. It is proven that the algorithm yields a sequence that globally converges to a critical point. As a result of some changes to the standard trust region method, namely a proximal regularisation of the trust region subproblem, it is shown that the local convergence rate is linear with an arbitrarily small ratio. Thus, convergen…
In this paper we develop random block coordinate gradient descent methods for minimizing large scale linearly constrained separable convex problems over networks. Since we have coupled constraints in the problem, we devise an algorithm that updates in parallel $τ\geq 2$ (block) components per iteration. Moreover, for this method the computations can be performed in a distributed fashion according to the structure of the network. However, its complexity per iteration is usually cheaper than of the full gradient method when the number of nodes $N$ in the network is large. We prove that for this …
We present an efficient algorithm for solving fractional programming problems whose objective functions are the ratio of a low-rank quadratic to a positive definite quadratic with convex constraints. The proposed algorithm for these convex-convex problems is based on the Shen-Yu Quadratic Transform which finds stationary points of concave-convex sum-of-ratios problems. We further use elements of the algorithm proposed in [arXiv:1802.10192] and the classic Dinkelbach approach to ensure convergence. We show that our algorithm performs better than previous algorithms for low-rank problems.
We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful proper…
We introduce a new technique for solving uni-parametric versions of linear programs, convex quadratic programs, and linear complementarity problems in which a single parameter is permitted to be present in any of the input data. We demonstrate the use of our method on a small, motivating example and present the results of a small number of computational tests demonstrating its utility for larger scale problems.
The Boosted Difference of Convex functions Algorithm (BDCA) has been recently introduced to accelerate the performance of the classical Difference of Convex functions Algorithm (DCA). This acceleration is achieved thanks to an extrapolation step from the point computed by DCA via a line search procedure. In this work, we propose an extension of BDCA that can be applied to difference of convex functions programs with linear constraints, and prove that every cluster point of the sequence generated by this algorithm is a Karush--Kuhn-Tucker point of the problem if the feasible set has a Slater po…
This paper proposes and analyzes a dampened proximal alternating direction method of multipliers (DP.ADMM) for solving linearly-constrained nonconvex optimization problems where the smooth part of the objective function is nonseparable. Each iteration of DP.ADMM consists of: (i) a sequence of partial proximal augmented Lagrangian (AL) updates, (ii) an under-relaxed Lagrange multiplier update, and (iii) a novel test to check whether the penalty parameter of the AL function should be updated. Under a basic Slater condition and some requirements related to the dampening factor and under-relaxatio…
For minimizing a strongly convex objective function subject to linear inequality constraints, we consider a penalty approach that allows one to utilize stochastic methods for problems with a large number of constraints and/or objective function terms. We provide upper bounds on the distance between the solutions to the original constrained problem and the penalty reformulations, guaranteeing the convergence of the proposed approach. We give a nested accelerated stochastic gradient method and propose a novel way for updating the smoothness parameter of the penalty function and the step-size. Th…
In this paper, we propose a mixed-binary convex quadratic programming reformulation for the box-constrained nonconvex quadratic integer program and then implement IBM ILOG CPLEX 12.6 to solve the new model. Computational results demonstrate that our approach clearly outperform the very recent state-of-the-art solvers.
The convex envelopes of the direct discrete measures, for the sparsity of vectors or for the low-rankness of matrices, have been utilized extensively as practical penalties in order to compute a globally optimal solution of the corresponding regularized least-squares models. Motivated mainly by the ideas in [Zhang'10, Selesnick'17, Yin, Parekh, Selesnick'19] to exploit nonconvex penalties in the regularized least-squares models without losing their overall convexities, this paper presents the Linearly involved Generalized Moreau Enhanced (LiGME) model as a unified extension of such utilization…
Mathematical programs with complementarity constraints are notoriously difficult to solve due to their nonconvexity and lack of constraint qualifications in every feasible point. This work focuses on the subclass of quadratic programs with linear complementarity constraints. A novel approach to solving a penalty reformulation using sequential convex programming and a homotopy on the penalty parameter is introduced. Linearizing the necessarily nonconvex penalty function yields convex quadratic subproblems, which have a constant Hessian matrix throughout all iterates. This allows solution comput…
In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with $\ell_1$-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex…
A lift-and-permute scheme of alternating direction method of multipliers (ADMM) is proposed for linearly constrained convex programming. It contains not only the newly developed balanced augmented Lagrangian method and its dual-primal variation, but also the proximal ADMM and Douglas-Rachford splitting algorithm. It helps to propose accelerated algorithms with worst-case $O(1/k^2)$ convergence rates in the case that the objective function to be minimized is strongly convex.
In this paper, we mainly study one class of convex mixed-integer nonlinear programming problems (MINLPs) with non-differentiable data. By dropping the differentiability assumption, we substitute gradients with subgradients obtained from KKT conditions, and use the outer approximation method to reformulate convex MINLP as one equivalent MILP master program. By solving a finite sequence of subproblems and relaxed MILP problems, we establish an outer approximation algorithm to find the optimal solution of this convex MINLP. The convergence of this algorithm is also presented. The work of this pap…
We study constrained stochastic programs where the decision vector at each time slot cannot be chosen freely but is tied to the realization of an underlying random state vector. The goal is to minimize a general objective function subject to linear constraints. A typical scenario where such programs appear is opportunistic scheduling over a network of time-varying channels, where the random state vector is the channel state observed, and the control vector is the transmission decision which depends on the current channel state. We consider a primal-dual type Frank-Wolfe algorithm that has a lo…
The charnes--cooper transformation and fractional programs with convex polynomials
Academy for Advanced Interdisciplinary Studies, Northeast Normal University（东北师范大学前沿交叉科学研究院）；Department of Applied Mathematics, Pukyong National University（釜山国立大学应用数学系）
This paper considers the problem of minimizing the time average of a stochastic process subject to time average constraints on other processes. A canonical example is minimizing average power in a data network subject to multi-user throughput constraints. Another example is a (static) convex program. Under a Slater condition, the drift-plus-penalty algorithm is known to provide an $O(ε)$ approximation to optimality with a convergence time of $O(1/ε^2)$. This paper proves the same result with a simpler technique and in a more general context that does not require the Slater condition. This pape…
We introduce an extension of Dual Dynamic Programming (DDP) to solve convex nonlinear dynamic programming equations. We call Inexact DDP (IDDP) this extension which applies to situations where some or all primal and dual subproblems to be solved along the iterations of the method are solved with a bounded error. We show that any accumulation point of the sequence of decisions is an approximate solution to the dynamic programming equations. When these errors tend to zero as the number of iterations goes to infinity, we show that IDDP solves the dynamic programming equations. We extend the analy…
In this paper we consider large-scale smooth optimization problems with multiple linear coupled constraints. Due to the non-separability of the constraints, arbitrary random sketching would not be guaranteed to work. Thus, we first investigate necessary and sufficient conditions for the sketch sampling to have well-defined algorithms. Based on these sampling conditions we developed new sketch descent methods for solving general smooth linearly constrained problems, in particular, random sketch descent and accelerated random sketch descent methods. From our knowledge, this is the first converge…
