Algoritmo do volume e otimização não diferenciável

We present a unifying framework for nonsmooth convex minimization bringing together $\epsilon$-subgradient algorithms and methods for the convex feasibility problem. This development is a natural step for $\epsilon$-subgradient methods in the direction of constrained optimization since the Euclidean projection frequently required in such methods is replaced by an approximate projection, which is often easier to compute. The developments are applied to incremental subgradient methods, resulting in new algorithms suitable to large-scale optimization problems, such as those arising in tomographic imaging.

We propose in this paper an algorithm for solving linearly constrained nondifferentiable convex programming problems. This algorithm combines the ideas of the affine scaling method with the subgradient method. It is a generalization of the dual and interior point method for min max problems proposed by J.F. Sturm and S. Zhang (1995), A dual and interior point approach to solve convex min–max problems, in D.-Z. Du and P.M. Pardalos (eds.)Minimax and Applications, pp. 69-78, Kluwer Academic Publishers. In the new method, the search direction is obtained by projecting in a scaled space a subgradient of the objective function with a logarithmic Barrier term. The stepsize choice is analogous to the stepsize choice in the usual subgradient method. Convergence of the method is established.

Many real-life scheduling, routing and location problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the so-called fixed linear crossing number problem (FLCNP). We show that this $\mathcal{NP}$-hard problem can be reduced to the well-known maximum cut problem. The latter problem was intensively studied in the literature; efficient exact algorithms based on the branch-and-cut technique have been developed. By an experimental evaluation on a variety of graphs, we show that using this reduction for solving FLCNP compares favorably to earlier branch-and-bound algorithms.

We consider combinatorial optimization problems with nonlinear objective functions. Solution approaches for this class of problems proposed so far are either highly problem-specific or they apply generic algorithms for constrained nonlinear optimization, which often does not yield satisfactory results in practice. Our aim is to develop, implement and experimentally evaluate exact algorithms that address the nonlinearity of the objective function and at the same time exploit the underlying combinatorial structure of the problem. To this end we follow two approaches. The first combines good polyhedral descriptions of the objective function and the feasible set in a branch and cut-algorithm. The second approach is based on Lagrangean decomposition. By decomposing the original problem into an unconstrained nonlinear problem and a linear combinatorial problem, we are able to compute strong dual bounds for the optimal value. The computation of lower bounds is then embedded into a branch and bound-algorithm. For many applications there already exist efficient algorithms for the combinatorial subproblem, thus an important aspect of this thesis is the study of the corresponding unconstrained nonlinear subproblems. Both approaches have the advantage that they can easily be adapted to a wide range of nonlinear combinatorial problems.We devise both polyhedral and decomposition- based algorithms for submodular applications from wireless network design and portfolio optimization and evaluate their performance experimentally. Exploiting the equivalence between unconstrained binary quadratic optimization and the maximum cut problem gives rise to a branch and cut-algorithm for quadratic combinatorial problems which we use to compute optimal layouts of tanglegrams, an application from computational biology. Additionally we study the effect of quadratic reformulation of linear constraints, both theoretically and experimentally. The last class of nonlinear combinatorial problems we consider are two-scenario problems. Here we propose a new technique to compute lower bounds in the unconstrained subproblem of the decomposition. Our computational study of the two-scenario minimum spanning tree problem shows that the new Lagrangean decomposition-based algorithm is able to solve significantly larger instances than the standard linearization approach.

To Jaynes, in his original paper [1], maxent is ‘a method of reasoning which ensures that no unconscious arbitrary assumptions have been introduced’, while fifty years later, the MAXENT conference home page suggests that the method ‘is not yet fully available to the statistics community at large’. In fact, it is possible to see that generalized maxent problems, often in disguise, do play a significant role in machine learning and statistics. Deviations from the classic form of the problem are typically used to incorporate some form of prior knowledge. Sometimes that knowledge would be difficult or impossible to represent with only linear constraints or an initial guess for the density.To clarify these connections, a good place to start is the classic maxent problem. This can then be generalized until the problem encompasses a large class of problems studied by the machine learning community. Relaxed constraints, generalizations of Shannon‐Boltzmann‐Gibbs (SBG) entropy and a few tools from convex analysis ...

CONTENTS Introduction Chapter I. The apparatus § 1. Convex analysis § 2. Approximations Chapter II. The family of functions ; the main conditions for a local minimum § 3. The problem of a conditional extremum and related concepts § 4. The main necessary condition for a local minimum § 5. The theorem about the estimation of the distance to the solution set of a system of strict non-linear inequalities and of an operator equation § 6. The main sufficient condition for a local minimum Chapter III. Conditions of order γ § 7. Necessary conditions of order γ § 8. Sufficient conditions of order γ Chapter IV. The family of functions § 9. Conditions for the equivalence of the families and § 10. A smooth system S Chapter V. Theory of a minimum on a set of sequences § 11. Main concepts and theorems of the general theory Chapter VI. Quadratic conditions for a minimum in various classes of extremal problems § 12. A smooth problem in Rn § 13. The problem of Bliss with end inequalities § 14. A vector optimization problem with a continuum of cost functions Supplement to Chapter VI Comments References

The capacitated multicommodity network flow problem presents itself in a number of problem contexts including transportation, communication, and production. To solve the large-scale multicommodity flow problems encountered in these fields, we develop dual-ascent heuristics and a primal solution generator. The dual-ascent solutions, in addition to determining lower bounds on the optimal objective function value, provide advanced starting solutions for use with primal-based solution techniques. The primal solution generator uses the dual-ascent solution to obtain heuristically primal solutions to the multicommodity flow problems. Computational experiments performed on three test problem sets show that the dual-ascent and primal heuristic procedures typically determine nearoptimal solutions quickly. In addition, by using the dual-ascent procedure to obtain advanced starting solutions, run times for optimal multicommodity flow procedures are reduced significantly and greatly improved solutions are obtained by the new primal solution generator. © 1993 John Wiley & Sons, Inc.

The capacitated multi-item lot sizing problem consists of finding a production schedule that minimizes over a finite number of periods the total production, holding inventory, and setup costs subject to demand and capacity constraints. The CLSP problem is NP-hard, while the problem of finding a feasible solution, which is polynomial if there are no set-up times, becomes NP-complete when set-up times are included. Approximate solutions can be obtained by heuristics. In this paper we consider an approach based on a Lagrangian relaxation of the capacity constraints. The relaxation is used in two ways. First, it generates a lower bound for the optimal value. Second, the primal and dual solutions of the relaxation (if available) are used to generate integer feasible solutions by primal or dual heuristics. We compare three methods of solving the Lagrangian relaxation: subgradient method, Kelley’s cutting plane method — also known as Dantzig-Wolfe decomposition — and the analytic center cutting plane method. We conclude that the analytic center cutting plane method performs as well, and sometimes better than subgradient optimization or Kelley’s cutting plane method.

Duality in robust linear regression using Huber's M-estimator

In this paper, we evaluate stochastic-computing simulated annealing (SC-SA) for solving large-scale combinato-rial optimization problems. SC-SA is designed using stochastic computing, where the computation is realized using random bitstream, resulting in fast converging to the global minimum energy of the problems. The proposed SC-SA is compared with a typical SA and existing simulated-annealing (SA) processors on the maximum cut (MAX-CUT) problems, such as Gset that is a benchmark for SA. The simulation results show that SC-SA realizes hundreds of times faster than a typical SA. In addition, SC-SA achieves better MAX-CUT scores than other existing methods on K2000 that is a complete 2000-node optimization problem.

CHAPTER 3 - Zero–One Implicit Enumeration

We propose a sparsity preserving algorithm for solving large-scale, nonlinear programming problems. The algorithm solves at each iteration a subproblem, which contains a linearized objective function augmented by a simple quadratic term and linearized constraints. The quadratic term added to the linearized objective function plays the role of step restriction which is essential in ensuring global convergence of the algorithm. If the conjugate gradient method or successive over-relaxation method is used to solve the subproblems, the sparsity of the original problem is preserved, because those methods only require simple operations on the rows of the constraint matrix. Thus, large-scale problems can be dealt with when the constraint matrices are sparse enough to be stored in a compact form. Practical implementation of the algorithm is described and computational results are reported.

The regulation of a decentralized large-scale linear system which consists of N subsystems S1, S2,…,SN interconnected with a subsystem S0 is investigated. Since in the state-space description the matrix describing the dynamics of this decentralized large-scale system consists of block elements depicting an arrow structure, the system is called a block arrow structure (BAS) decentralized large-scale system and its regulation problem is called a BAS-decentralized large-scale linear regulator problem. In this paper, the time-invariant BAS-decentralized large-scale linear regulator problem is mathematically formulated and a solution to this problem is given by developing a suboptimal linear state feedback controller with the features of (a) partial decentralization (utilizing the interconnections), (b) parallel implementation, and (c) structure flexibility so that the addition and/or deletion of any subsystems within the BAS model specifications does not require the solution of the problem from the beginning ...

This paper is concerned with the long-standing problem of optimal unit commitment in an electric power system. We follow the traditional formulation of this problem which gives rise to a large-scale, dynamic, mixed-integer programming problem. We describe a solution methodology based on duality, Lagrangian relaxation, and nondifferentiable optimization that has two unique features. First, computational requirements typically grow only linearly with the number of generating units. Second, the duality gap decreases in relative terms as the number of units increases, and as a result our algorithm tends to actually perform better for problems of large size. This allows for the first time consistently reliable solution of large practical problems involving several hundreds of units within realistic time constraints. Aside from the unit commitment problem, this methodology is applicable to a broad class of large-scale dynamic scheduling and resource allocation problems involving integer variables.

We provide experimental evidence that current desktop computers feature enough computational power to solve large-scale dense linear algebra problems. While the high computational cost of the numerical methods for solving these problems can be tackled by the multiple cores of current processors, we propose to use the disk to store the large data structures associated with these applications. Our results also show that the limited amount of RAM and the comparatively slow disk of the system pose no problem for the solution of very large dense linear systems and linear least-squares problems. Thus, current desktop computers are revealed as an appealing, cost-effective platform for research groups that have to deal with large dense linear algebra problems but have no direct access to large computing facilities.