Optimization problems over permutation matrices appear widely in facility layout, chip design, scheduling, pattern recognition, computer vision, graph matching, etc. Since this problem is NP-hard due to the combinatorial nature of permutation matrices, we relax the variable to be the more tractable doubly stochastic matrices and add an @math -norm ( @math ) regularization term to the objective function. The optimal solutions of the @math -regularized problem are the same as the original problem if the regularization parameter is sufficiently large. A lower bound estimation of the nonzero entries of the stationary points and some connections between the local minimizers and the permutation matrices are further established. Then we propose an @math regularization algorithm with local refinements. The algorithm approximately solves a sequence of @math regularization subproblems by the projected gradient method using a nonmontone line search with the Barzilai-Borwein step sizes. Its performance can be further improved if it is combined with certain local search methods, the cutting plane techniques as well as a new negative proximal point scheme. Extensive numerical results on QAPLIB and the bandwidth minimization problem show that our proposed algorithms can often find reasonably high quality solutions within a competitive amount of time.
The aforementioned regularization or relaxation methods all considered solving optimization over @math . By using the fact that @math , Wen and Yin @cite_41 reformulated QAP as X ^ n n , (A^ (X X) B (X X)^ ) X^ X = I_n, X 0 and proposed an efficient algorithm by applying the augmented Lagrangian method to handle the constraint @math . In their algorithm, a sequence of augmented Lagrangian subproblems with orthogonality constraint is solved inexactly.
@cite_41
2277973662 2346942497 2086953401 2473549844
This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints: @math minimizef1(x1)+ź+fN(xN)subject toA1x1+ź+ANxN=c,x1źX1,ź,xNźXN,where @math Nź2, @math fi are convex functions, @math Ai are matrices, and @math Xi are feasible sets for variable @math xi. Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices @math Ai are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices @math Ai). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that @math źxk+1-xkźM2 converges at a rate of o(1 k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported. TAO HUANG: Continuous Optimization Methods for the Quadratic Assignment Problem (Under the direction of Jon W. Tolle) In this dissertation we have studied continuous optimization techniques as they are applied in nonlinear 0-1 programming. Specifically, the methods of relaxation with a penalty function have been carefully investigated. When the strong equivalence properties hold, we are guaranteed an integer solution to the original 0-1 problem. The quadratic assignment problem (QAP) possesses such properties and consequently we have developed an algorithm for the QAP based on the method of relaxation using the quadratic penalty function. In our algorithm we have applied two pre-conditioning techniques that enables us to devise a scheme to find a good initial point and hence obtain good solutions to the QAP. Furthermore, we have shown how quadratic cuts can be used to improve on the current solutions. Extensive numerical results on several sets of QAP test problems (including the QAPLIB) have been reported and these results show our algorithm produces good solutions for certain classes of problems in a small amount of time. Minimization with orthogonality constraints (e.g., (X^ X = I )) and or spherical constraints (e.g., ( x _2 = 1 )) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we apply the Cayley transform—a Crank-Nicolson-like update scheme—to preserve the constraints and based on it, develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842 to the best known solution on the largest problem “tai256c” in QAPLIB can be reached in 5 min on a typical laptop. We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math .