The nuclear norm is widely used to induce low-rank solutions for many optimization problems with matrix variables. Recently, it has been shown that the augmented Lagrangian method (ALM) and the alternating direction method (ADM) are very efficient for many convex programming problems arising from various applications, provided that the resulting subproblems are sufficiently simple to have closed-form solutions. In this paper, we are interested in the application of the ALM and the ADM for some nuclear norm involved minimization problems. When the resulting subproblems do not have closed-form solutions, we propose to linearize these subproblems such that closed-form solutions of these linearized subproblems can be easily derived. Global convergence results of these linearized ALM and ADM are established under standard assumptions. Finally, we verify the effectiveness and efficiency of these new methods by some numerical experiments.
In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, φ(x0, . . . , x p, y), subject to coupled linear equality constraints. Our ADMM updates
each of the primal variables x0, . . . , x p, y, followed by updating the dual variable.We separate the variable y from xi ’s as it has a special role in our analysis. The developed convergence guarantee covers a variety of nonconvex functions such as piecewise linear functions, lq quasi-norm, Schatten-q quasi-norm (0 < q < 1), minimax concave penalty (MCP), and smoothly clipped absolute deviation penalty. It also allows nonconvex constraints such as compact manifolds (e.g., spherical, Stiefel, and Grassman manifolds) and linear complementarity constraints. Also, the x0-block can be almost any lower semi-continuous function. By applying our analysis, we show, for the first time, that several ADMM algorithms applied to solve nonconvex models in statistical learning, optimization on manifold, and matrix decomposition are guaranteed to converge. Our results provide sufficient conditions for ADMM to converge on (convex or nonconvex) monotropic programs with three or more blocks, as
they are special cases of our model. ADMM has been regarded as a variant to the augmented Lagrangian method (ALM). We present a simple example to illustrate how ADMM converges but ALM diverges with bounded penalty parameter β. Indicated by this example and other analysis in this paper, ADMM might be a better choice than ALM for some nonconvex nonsmooth problems, because ADMM is not only easier to implement, it is also more likely to converge for the concerned scenarios.
In this paper, we analyze the minimization of seminorms ∥L · ∥ on R^n under
the constraint of a bounded I-divergence D(b,H·) for rather general linear
operators H and L. The I-divergence is also known as Kullback–Leibler
divergence and appears in many models in imaging science, in particular when
dealing with Poisson data but also in the case of multiplicative Gamma noise.
Often H represents, e.g., a linear blur operator and L is some discrete derivative
or frame analysis operator. A central part of this paper consists in proving
relations between the parameters of I-divergence constrained and penalized
problems. To solve the I-divergence constrained problem, we consider various
first-order primal–dual algorithms which reduce the problem to the solution of
certain proximal minimization problems in each iteration step. One of these
proximation problems is an I-divergence constrained least-squares problem
which can be solved based on Morozov’s discrepancy principle by a Newton
method. We prove that these algorithms produce not only a sequence of
vectors which converges to a minimizer of the constrained problem but also
a sequence of parameters which converges to a regularization parameter so
that the corresponding penalized problem has the same solution. Furthermore,
we derive a rule for automatically setting the constraint parameter for data
corrupted by multiplicative Gamma noise. The performance of the various
algorithms is finally demonstrated for different image restoration tasks both for
images corrupted by Poisson noise and multiplicative Gamma noise.
Aust G, Buscher U. Vertical Cooperative Advertising and Pricing Decisions in a Manufacturer-Retailer Supply Chain: A Game-Theoretic Approach[J]. European Journal of Operational Research, 2012, 223(2): 473-482.
This paper focuses on pricing and vertical cooperative advertising decisions in a two-tier supply chain. Using a
Stackelberg game model where the manufacturer acts as the game leader and the retailer acts as the game follower, we
obtain closed-form equilibrium solution and explicitly show how pricing and advertising decisions are made. When
market demand decreases exponentially with respect to the retail price and increases with respect to national and local
advertising expenditures in an additive way, the manufacturer benefits from providing percentage reimbursement for the
retailer’s local advertising expenditure when demand price elasticity is large enough. Whether the manufacturer benefits
from cooperative advertising is also closely related to supply chain member’s relative advertising efficiency. In the
decision for adopting coop advertising strategy, it is critical for the manufacturer to identify how market demand depends
on national and local advertisements. The findings from this research can enhance our understanding of cooperative
advertising decisions in a two-tier supply chain with price-dependent demand.
We introduce a parallel machine scheduling problem in
which the processing times of jobs are not given in advance but are
determined by a system of linear constraints. The objective is to
minimize the makespan, i.e., the maximum job completion time among
all feasible choices. This novel problem is motivated by various
real-world application scenarios. We discuss the computational
complexity and algorithms for various settings of this problem. In
particular, we show that if there is only one machine with an
arbitrary number of linear constraints, or there is an arbitrary
number of machines with no more than two linear constraints, or both
the number of machines and the number of linear constraints are
fixed constants, then the problem is polynomial-time solvable via
solving a series of linear programming problems. If both the number
of machines and the number of constraints are inputs of the problem
instance, then the problem is NP-Hard. We further propose several
approximation algorithms for the latter case.
In this paper, we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a Hilbert space. Our results are achieved under a bounded Holder regularity assumption which generalizes the well-known notion of bounded linear regularity. As an application of our results, we provide a convergence rate analysis for many important iterative methods in solving broad mathematical problems such as convex feasibility problems and variational inequality problems. These include Krasnoselskii–Mann iterations, the cyclic projection algorithm, forward-backward splitting and the Douglas–Rachford feasibility algorithm along with some variants. In the important case in which the underlying sets are convex sets described by convex polynomials in a finite dimensional space, we show that the Holder regularity properties are automatically satisfied,
from which sublinear convergence follows.
This paper studies the capacity allocation game between duopolistic airlines which could offer callable products. Previous literature has shown that callable products provide a riskless source of additional rev- enue for a monopolistic airline. We examine the impact of the introduction of callable products on the revenues and the booking limits of duopolistic airlines. The analytical results demonstrate that, when there is no spill of low-fare customers, offering callable products is a dominant strategy of both airlines and provides Pareto gains to both airlines. When customers of both fare classes spill, offering callable products is no longer a dominant strategy and may harm the revenues of the airlines. Numerical ex- amples demonstrate that whether the two airlines offer callable products and whether offering callable products is beneficial to the two airlines mainly depends on their loads and capacities. Specifically, when the difference between the loads of the airlines is large, the loads of the airlines play the most important role. When the difference between the loads of the airlines is small, the capacities of the airlines play the most important role. Moreover, numerical examples show that the booking limits of the two airlines in the case with callable products are always higher than those in the case without callable products.