Mixed-Integer Linear Programming (MILP) plays an important role across a range of scientific disciplines and within areas of strategic importance to society. The MILP problems, however, suffer from combinatorial complexity. Because of integer decision variables, as the problem size increases, the number of possible solutions increases super-linearly thereby leading to a drastic increase in the computational effort. To efficiently solve MILP problems, a "price-based" decomposition and coordination approach is developed to exploit 1. the super-linear reduction of complexity upon the decomposition and 2. the geometric convergence potential inherent to Polyak's stepsizing formula for the fastest coordination possible to obtain near-optimal solutions in a computationally efficient manner. Unlike all previous methods to set stepsizes heuristically by adjusting hyperparameters, the key novel way to obtain stepsizes is purely decision-based: a novel "auxiliary" constraint satisfaction problem is solved, from which the appropriate stepsizes are inferred. Testing results for large-scale Generalized Assignment Problems (GAP) demonstrate that for the majority of instances, certifiably optimal solutions are obtained. For stochastic job-shop scheduling as well as for pharmaceutical scheduling, computational results demonstrate the two orders of magnitude speedup as compared to Branch-and-Cut (B&C). The new method has a major impact on the efficient resolution of complex Mixed-Integer Programming (MIP) problems arising within a variety of scientific fields.
This paper explores the relationship between numerical integrators and optimal control algorithms. Specifically, the performance of the differential dynamical programming (DDP) algorithm is examined when a variational integrator and a newly proposed surrogate variational integrator are used to propagate and linearize system dynamics. Surrogate variational integrators, derived from backward error analysis, achieve higher levels of accuracy while maintaining the same integration complexity as nominal variational integrators. The increase in the integration accuracy is shown to have a large effec…
In the context of augmented Lagrangian approaches for solving semidefinite programming problems, we investigate the possibility of eliminating the positive semidefinite constraint on the dual matrix by employing a factorization. Hints on how to deal with the resulting unconstrained maximization of the augmented Lagrangian are given. We further use the approximate maximum of the augmented Lagrangian with the aim of improving the convergence rate of alternating direction augmented Lagrangian frameworks. Numerical results are reported, showing the benefits of the approach.
In this paper, we study augmented Lagrangian functions for nonlinear semidefinite programming (NSDP) problems with exactness properties. The term exact is used in the sense that the penalty parameter can be taken appropriately, so a single minimization of the augmented Lagrangian recovers a solution of the original problem. This leads to reformulations of NSDP problems into unconstrained nonlinear programming ones. Here, we first establish a unified framework for constructing these exact functions, generalizing Di Pillo and Lucidi's work from 1996, that was aimed at solving nonlinear programmi…
The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global epsilon-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of const…
Mixed integer quadratic programming (MIQP) is the problem of minimizing a convex quadratic function over mixed integer points in a rational polyhedron. This paper focuses on the augmented Lagrangian dual (ALD) for MIQP. ALD augments the usual Lagrangian dual with a weighted nonlinear penalty on the dualized constraints. We first prove that ALD will reach a zero duality gap asymptotically as the weight on the penalty goes to infinity under some mild conditions on the penalty function. We next show that a finite penalty weight is enough for a zero gap when we use any norm as the penalty function…
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…
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 …
We develop a decomposition method based on the augmented Lagrangian framework to solve a broad family of semidefinite programming problems, possibly with nonlinear objective functions, nonsmooth regularization, and general linear equality/inequality constraints. In particular, the positive semidefinite variable along with a group of linear constraints can be transformed into a variable on a smooth manifold via matrix factorization. The nonsmooth regularization and other general linear constraints are handled by the augmented Lagrangian method. Therefore, each subproblem can be solved by a semi…
A sparse linear programming (SLP) problem is a linear programming problem equipped with a sparsity (or cardinality) constraint, which is nonconvex and discontinuous theoretically and generally NP-hard computationally due to the combinatorial property involved. By rewriting the sparsity constraint into a disjunctive form, we present an explicit formula of its Lagrangian dual in terms of an unconstrained piecewise-linear convex programming problem which admits a strong duality. A semi-proximal alternating direction method of multipliers (sPADMM) is then proposed to solve this dual problem by tak…
Due to the possible lack of primal-dual-type error bounds, the superlinear convergence for the Karush-Kuhn-Tucker (KKT) residues of the sequence generated by augmented Lagrangian method (ALM) for solving convex composite conic programming (CCCP) has long been an outstanding open question. In this paper, we aim to resolve this issue by first conducting convergence rate analysis for the ALM with Rockafellar's stopping criteria under only a mild quadratic growth condition on the dual of CCCP. More importantly, by further assuming that the Robinson constraint qualification holds, we establish the …
In this paper, we consider the linearly constrained composite convex optimization problem, whose objective is a sum of a smooth function and a possibly nonsmooth function. We propose an inexact augmented Lagrangian (IAL) framework for solving the problem. The stopping criterion used in solving the augmented Lagrangian (AL) subproblem in the proposed IAL framework is weaker and potentially much easier to check than the one used in most of the existing IAL frameworks/methods. We analyze the global convergence and the non-ergodic convergence rate of the proposed IAL framework.
Optimal distribution of power among generating units to meet a specific demand subject to system constraints is an ongoing research topic in the power system community. The problem, even in a static setting, turns out to be hard to solve with conventional optimization methods owing to the consideration of valve-point effects which make the cost function nonsmooth and nonconvex. This difficulty gave rise to the proliferation of population-based global heuristics in order to address the multi-extremal and nonsmooth problem. In this paper, we address the economic load dispatch problem (ELDP) with…
We consider the convex minimization model with both linear equality and inequality constraints, and reshape the classic augmented Lagrangian method (ALM) by balancing its subproblems. As a result, one of its subproblems decouples the objective function and the coefficient matrix without any extra condition, and the other subproblem becomes a positive definite system of linear equations or a positive definite linear complementary problem. The balanced ALM advances the classic ALM by enlarging its applicable range, balancing its subproblems, and improving its implementation. We also extend our d…
We propose a semi-proximal augmented Lagrangian based decomposition method for convex composite quadratic conic programming problems with primal block angular structures. Using our algorithmic framework, we are able to naturally derive several well known augmented Lagrangian based decomposition methods for stochastic programming such as the diagonal quadratic approximation method of Mulvey and Ruszczyński. Moreover, we are able to derive novel enhancements and generalizations of these well known methods. We also propose a semi-proximal symmetric Gauss-Seidel based alternating direction method …
We present a new primal-dual algorithm for computing the value of the Lagrangian dual of a stochastic mixed-integer program (SMIP) formed by relaxing its nonanticipativity constraints. This dual is widely used in decomposition methods for the solution of SMIPs. The algorithm relies on the well-known progressive hedging method, but unlike previous progressive hedging approaches for SMIP, our algorithm can be shown to converge to the optimal Lagrangian dual value. The key improvement in the new algorithm is an inner loop of optimized linearization steps, similar to those taken in the classical F…
We design inexact proximal augmented Lagrangian based decomposition methods for convex composite programming problems with dual block-angular structures. Our methods are particularly well suited for convex quadratic programming problems arising from stochastic programming models. The algorithmic framework is based on the application of the abstract inexact proximal ADMM framework developed in [Chen, Sun, Toh, Math. Prog. 161:237--270] to the dual of the target problem, as well as the application of the recently developed symmetric Gauss-Seidel decomposition theorem for solving a proximal multi…
Solution Seeker AS；Norwegian University of Science and Technology（挪威科技大学）
We consider the embedding of piecewise-linear deep neural networks (ReLU networks) as surrogate models in mixed-integer linear programming (MILP) problems. A MILP formulation of ReLU networks has recently been applied by many authors to probe for various model properties subject to input bounds. The formulation is obtained by programming each ReLU operator with a binary variable and applying the big-M method. The efficiency of the formulation hinges on the tightness of the bounds defined by the big-M values. When ReLU networks are embedded in a larger optimization problem, the presence of outp…
For years, there has been interest in approximation methods for solving dynamic programming problems, because of the inherent complexity in computing optimal solutions characterized by Bellman's principle of optimality. A wide range of approximate dynamic programming (ADP) methods now exists. It is of great interest to guarantee that the performance of an ADP scheme be at least some known fraction, say $β$, of optimal. This paper introduces a general approach to bounding the performance of ADP methods, in this sense, in the stochastic setting. The approach is based on new results for bounding …
Augmented Lagrangian methods for nonlinear semidefinite programming with complementarity constraints
Federal University of São Paulo（圣保罗联邦大学）；State University of Maringá（马林加州立大学）；Federal University of Paraná（巴拉那联邦大学）；Instituto Federal de Santa Catarina (IFSC)（圣卡塔琳娜联邦学院）
The augmented Lagrangian method (ALM) is classic for canonical convex programming problems with linear constraints, and it finds many applications in various scientific computing areas. A major advantage of the ALM is that the step for updating the dual variable can be further relaxed with a step size in $(0,2)$, and this advantage can easily lead to numerical acceleration for the ALM. When a separable convex programming problem is discussed and a corresponding splitting version of the classic ALM is considered, convergence may not be guaranteed and thus it is seemingly impossible that a step …
In this paper, we aim to solve high dimensional convex quadratic programming (QP) problems with a large number of quadratic terms, linear equality and inequality constraints. In order to solve the targeted {\bf QP} problems to a desired accuracy efficiently, we develop a two-phase {\bf P}roximal {\bf A}ugmented {\bf L}agrangian method {(QPPAL)}, with Phase I to generate a reasonably good initial point to warm start Phase II to obtain an accurate solution efficiently. More specifically, in Phase I, based on the recently developed symmetric Gauss-Seidel (sGS) decomposition technique, we design a…
In this paper, we present a majorized semismooth Newton-CG augmented Lagrangian method, called SDPNAL$+$, for semidefinite programming (SDP) with partial or full nonnegative constraints on the matrix variable. SDPNAL$+$ is a much enhanced version of SDPNAL introduced by Zhao, Sun and Toh [SIAM Journal on Optimization, 20 (2010), pp.~1737--1765] for solving generic SDPs. SDPNAL works very efficiently for nondegenerate SDPs but may encounter numerical difficulty for degenerate ones. Here we tackle this numerical difficulty by employing a majorized semismooth Newton-CG augmented Lagrangian method…
We present the implementation of a trust-region Newton algorithm ExaTron for bound-constrained nonlinear programming problems, fully running on multiple GPUs. Without data transfers between CPU and GPU, our implementation has achieved the elimination of a major performance bottleneck under a memory-bound situation, particularly when solving many small problems in batch. We discuss the design principles and implementation details for our kernel function and core operations. Different design choices are justified by numerical experiments. By using the application of distributed control of altern…
The unit commitment problem (UC) is crucial for the operation and market mechanism of power systems. With the development of modern electricity, the scale of power systems is expanding, and solving the UC problem is also becoming more and more difficult. To this end, this paper proposes a new fast solution method based on Lagrangian relaxation and dynamic program-ming. Firstly, the UC solution is estimated to be an initial trial UC solution by a fast method based on Lagrangian relaxation. This initial trial UC solution fully considers the system-wide con-straints. Secondly, a dynamic programmi…
In this chapter we derive computational complexity certifications of first order inexact dual methods for solving general smooth constrained convex problems which can arise in real-time applications, such as model predictive control. When it is difficult to project on the primal constraint set described by a collection of general convex functions, we use the Lagrangian relaxation to handle the complicated constraints and then, we apply dual (fast) gradient algorithms based on inexact dual gradient information for solving the corresponding dual problem. The iteration complexity analysis is base…
In this paper we present a complete iteration complexity analysis of inexact first order Lagrangian and penalty methods for solving cone constrained convex problems that have or may not have optimal Lagrange multipliers that close the duality gap. We first assume the existence of optimal Lagrange multipliers and study primal-dual first order methods based on inexact information and augmented Lagrangian smoothing or Nesterov type smoothing. For inexact (fast) gradient augmented Lagrangian methods we derive a total computational complexity of $\mathcal{O}\left(\frac{1}ε\right)$ projections onto …
First-order methods have been popularly used for solving large-scale problems. However, many existing works only consider unconstrained problems or those with simple constraint. In this paper, we develop two first-order methods for constrained convex programs, for which the constraint set is represented by affine equations and smooth nonlinear inequalities. Both methods are based on the classic augmented Lagrangian function. They update the multipliers in the same way as the augmented Lagrangian method (ALM) but employ different primal variable updates. The first method, at each iteration, per…
We propose a new first-order augmented Lagrangian algorithm ALCC for solving convex conic programs of the form min{rho(x)+gamma(x): Ax-b in K, x in chi}, where rho and gamma are closed convex functions, and gamma has a Lipschitz continuous gradient, A is mxn real matrix, K is a closed convex cone, and chi is a "simple" convex compact set such that optimization problems of the form min{rho(x)+|x-x0|_2^2: x in chi} can be efficiently solved for any given x0. We show that any limit point of the primal ALCC iterates is an optimal solution of the conic convex problem, and the dual ALCC iterates hav…
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…
In this paper, we present a two-phase augmented Lagrangian method, called QSDPNAL, for solving convex quadratic semidefinite programming (QSDP) problems with constraints consisting of a large number of linear equality, inequality constraints, a simple convex polyhedral set constraint, and a positive semidefinite cone constraint. A first order algorithm which relies on the inexact Schur complement based decomposition technique is developed in QSDPNAL-Phase I with the aim of solving a QSDP problem to moderate accuracy or using it to generate a reasonably good initial point for the second phase. …
In this paper we consider a class of convex conic programming. In particular, we first propose an inexact augmented Lagrangian (I-AL) method that resembles the classical I-AL method for solving this problem, in which the augmented Lagrangian subproblems are solved approximately by a variant of Nesterov's optimal first-order method. We show that the total number of first-order iterations of the proposed I-AL method for finding an $ε$-KKT solution is at most $\mathcal{O}(ε^{-7/4})$. We then propose an adaptively regularized I-AL method and show that it achieves a first-order iteration complexity…
The augmented Lagrangian method (ALM) is a benchmark for convex programming problems with linear constraints; ALM and its variants for linearly equality-constrained convex minimization models have been well studied in the literature. However, much less attention has been paid to ALM for efficiently solving linearly inequality-constrained convex minimization models. In this paper, we exploit an enlightening reformulation of the newly developed indefinite linearized ALM for the equality-constrained convex optimization problem, and present a new indefinite linearized ALM scheme for efficiently so…
We consider semi-infinite linear programs with countably many constraints indexed by the natural numbers. When the constraint space is the vector space of all real valued sequences, we show the finite support (Haar) dual is equivalent to the algebraic Lagrangian dual of the linear program. This settles a question left open by Anderson and Nash [Linear programming in infinite dimensional spaces : theory and applications, Wiley 1987]. This result implies that if there is a duality gap between the primal linear program and its finite support dual, then this duality gap cannot be closed by conside…
Solving large scale convex semidefinite programming (SDP) problems has long been a challenging task numerically. Fortunately, several powerful solvers including SDPNAL, SDPNAL+ and QSDPNAL have recently been developed to solve linear and convex quadratic SDP problems to high accuracy successfully. These solvers are based on the augmented Lagrangian method (ALM) applied to the dual problems with the subproblems being solved by semismooth Newton-CG methods. Noticeably, thanks to Rockafellar's general theory on the proximal point algorithms, the primal iteration sequence generated by the ALM enjo…
We consider solving nonlinear optimization problems with a stochastic objective and deterministic equality constraints. We assume for the objective that its evaluation, gradient, and Hessian are inaccessible, while one can compute their stochastic estimates by, for example, subsampling. We propose a stochastic algorithm based on sequential quadratic programming (SQP) that uses a differentiable exact augmented Lagrangian as the merit function. To motivate our algorithm design, we first revisit and simplify an old SQP method \citep{Lucidi1990Recursive} developed for solving deterministic problem…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Microsoft Research（微软研究院）
Prediction+optimization is a common real-world paradigm where we have to predict problem parameters before solving the optimization problem. However, the criteria by which the prediction model is trained are often inconsistent with the goal of the downstream optimization problem. Recently, decision-focused prediction approaches, such as SPO+ and direct optimization, have been proposed to fill this gap. However, they cannot directly handle the soft constraints with the $max$ operator required in many real-world objectives. This paper proposes a novel analytically differentiable surrogate object…
In this paper, we adopt the augmented Lagrangian method (ALM) to solve convex quadratic second-order cone programming problems (SOCPs). Fruitful results on the efficiency of the ALM have been established in the literature. Recently, it has been shown in [Cui, Sun, and Toh, {\em Math. Program.}, 178 (2019), pp. 381--415] that if the quadratic growth condition holds at an optimal solution for the dual problem, then the KKT residual converges to zero R-superlinearly when the ALM is applied to the primal problem. Moreover, Cui, Ding, and Zhao [{\em SIAM J. Optim.}, 27 (2017), pp. 2332-2355] provid…
Several well known large scale linear programming decomposition methodologies exist. Benders Decomposition, which covers the case where some small subset of variables link the otherwise separable subproblems. Dantzig-Wolfe decomposition and Lagrangian decompositions, which cover the case where some few constraints link the otherwise separable subproblems, and finally the "Cross-Decomposition" originating from TJ Van Roy which enables one to deal with both linking constraints and linking variables by essentially alternating iteratively between the Benders and the Lagrangian Decomposition. In th…
We consider a multi-objective risk-averse two-stage stochastic programming problem with a multivariate convex risk measure. We suggest a convex vector optimization formulation with set-valued constraints and propose an extended version of Benson's algorithm to solve this problem. Using Lagrangian duality, we develop scenario-wise decomposition methods to solve the two scalarization problems appearing in Benson's algorithm. Then, we propose a procedure to recover the primal solutions of these scalarization problems from the solutions of their Lagrangian dual problems. Finally, we test our algor…
The sparse nonlinear programming (SNP) problem has wide applications in signal and image processing, machine learning, pattern recognition, finance and management, etc. However, the computational challenge posed by SNP has not yet been well resolved due to the nonconvex and discontinuous $\ell_0$-norm involved. In this paper, we resolve this numerical challenge by developing a fast Newton-type algorithm. As a theoretical cornerstone, we establish a first-order optimality condition for SNP based on the concept of strong $β$-Lagrangian stationarity via the Lagrangian function, and reformulate it…
A novel algorithm to solve the quadratic programming problem over ellipsoids is proposed. This is achieved by splitting the problem into two optimisation sub-problems, quadratic programming over a sphere and orthogonal projection. Next, an augmented-Lagrangian algorithm is developed for this multiple constraint optimisation. Benefit from the fact that the QP over a single sphere can be solved in a closed form by solving a secular equation, we derive a tighter bound of the minimiser of the secular equation. We also propose to generate a new psd matrix with a low condition number from the matric…
