Journal of Global Optimization最新文献

Abstract

This paper presents an improved method for computing convex and concave relaxations of the parametric solutions of ordinary differential equations (ODEs). These are called state relaxations and are crucial for solving dynamic optimization problems to global optimality via branch-and-bound (B &B). The new method improves upon an existing approach known as relaxation preserving dynamics (RPD). RPD is generally considered to be among the best available methods for computing state relaxations in terms of both efficiency and accuracy. However, it requires the solution of a hybrid dynamical system, whereas other similar methods only require the solution of a simple system of ODEs. This is problematic in the context of branch-and-bound because it leads to higher cost and reduced reliability (i.e., invalid relaxations can result if hybrid mode switches are not detected numerically). Moreover, there is no known sensitivity theory for the RPD hybrid system. This makes it impossible to compute subgradients of the RPD relaxations, which are essential for efficiently solving the associated B &B lower bounding problems. To address these limitations, this paper presents a small but important modification of the RPD theory, and a corresponding modification of its numerical implementation, that crucially allows state relaxations to be computed by solving a system of ODEs rather than a hybrid system. This new RPD method is then compared to the original using two examples and shown to be more efficient, more robust, and of almost identical accuracy.

We present a novel relaxation for general nonconvex sparse MINLP problems, called overlapping convex hull relaxation (CHR). It is defined by replacing all nonlinear constraint sets by their convex hulls. If the convex hulls are disjunctive, e.g. if the MINLP is block-separable, the CHR is equivalent to the convex hull relaxation obtained by (standard) column generation (CG). The CHR can be used for computing an initial lower bound in the root node of a branch-and-bound algorithm, or for computing a start vector for a local-search-based MINLP heuristic. We describe a dynamic block and column generation (DBCG) MINLP algorithm to generate the CHR by dynamically adding aggregated blocks. The idea of adding aggregated blocks in the CHR is similar to the well-known cutting plane approach. Numerical experiments on nonconvex MINLP instances show that the duality gap can be significantly reduced with the results of CHRs. DBCG is implemented as part of the CG-MINLP framework Decogo, see https://decogo.readthedocs.io/en/latest/index.html.

In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in ({{,mathrm{mathcal {P}},}}cap {{,mathrm{mathbb {Z}},}}^n), assuming that ({{,mathrm{mathcal {P}},}}) is a polyhedron, defined by systems (A x le b) or (Ax = b,, x ge 0) with a sparse matrix A. We develop algorithms for these problems that outperform state-of-the-art ILP and counting algorithms on sparse instances with bounded elements in terms of the computational complexity. Assuming that the matrix A has bounded elements, our complexity bounds have the form (s^{O(n)}), where s is the minimum between numbers of non-zeroes in columns and rows of A, respectively. For (s = obigl (log n bigr )), this bound outperforms the state-of-the-art ILP feasibility complexity bound ((log n)^{O(n)}), due to Reis & Rothvoss (in: 2023 IEEE 64th Annual symposium on foundations of computer science (FOCS), IEEE, pp. 974–988). For (s = phi ^{o(log n)}), where (phi ) denotes the input bit-encoding length, it outperforms the state-of-the-art ILP counting complexity bound (phi ^{O(n log n)}), due to Barvinok et al. (in: Proceedings of 1993 IEEE 34th annual foundations of computer science, pp. 566–572, https://doi.org/10.1109/SFCS.1993.366830, 1993), Dyer, Kannan (Math Oper Res 22(3):545–549, https://doi.org/10.1287/moor.22.3.545, 1997), Barvinok, Pommersheim (Algebr Combin 38:91–147, 1999), Barvinok (in: European Mathematical Society, ETH-Zentrum, Zurich, 2008). We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximum Matching problems.

The aim of this paper is to introduce a nonlinear scalarization function in set optimization based on the concept of null set which was introduced by Wu (J Math Anal Appl 472(2):1741–1761, 2019). We introduce a notion of pseudo algebraic interior of a set and define a weak set order relation using the concept of null set. We investigate several properties of this nonlinear scalarization function. Further, we characterize the set order relations and investigate optimality conditions for solution sets in set optimization based on the concept of null set. Finally, a numerical example is provided to compute a weak minimal solution using this nonlinear scalarization function.

In this paper, we propose an inertial alternating direction method of multipliers for solving a class of non-convex multi-block optimization problems with nonlinear coupling constraints. Distinctive features of our proposed method, when compared with other alternating direction methods of multipliers for solving non-convex problems with nonlinear coupling constraints, include: (i) we apply the inertial technique to the update of primal variables and (ii) we apply a non-standard update rule for the multiplier by scaling the multiplier by a factor before moving along the ascent direction where a relaxation parameter is allowed. Subsequential convergence and global convergence are presented for the proposed algorithm.

Trust-region methods have received massive attention in a variety of continuous optimization. They aim to obtain a trial step by minimizing a quadratic model in a region of a certain trust-region radius around the current iterate. This paper proposes an adaptive Riemannian trust-region algorithm for optimization on manifolds, in which the trust-region radius depends linearly on the norm of the Riemannian gradient at each iteration. Under mild assumptions, we establish the liminf-type convergence, lim-type convergence, and global convergence results of the proposed algorithm. In addition, the proposed algorithm is shown to reach the conclusion that the norm of the Riemannian gradient is smaller than (epsilon ) within ({mathcal {O}}(frac{1}{epsilon ^2})) iterations. Some numerical examples of tensor approximations are carried out to reveal the performances of the proposed algorithm compared to the classical Riemannian trust-region algorithm.

This paper proposes two simple and elegant proximal-type algorithms to solve equilibrium problems with pseudo-monotone bifunctions in the setting of Hilbert spaces. The proposed algorithms use one proximal point evaluation of the bifunction at each iteration. Consequently, prove that the sequences of iterates generated by the first algorithm converge weakly to a solution of the equilibrium problem (assuming existence) and obtain a linear convergence rate under standard assumptions. We also design a viscosity version of the first algorithm and obtain its corresponding strong convergence result. Some popular existing algorithms in the literature are recovered. We finally give some numerical tests and compare our algorithms with some related ones to show the performance and efficiency of our proposed algorithms.

In this paper, we consider the problem that minimizing the sum of a nonsmooth function with a smooth one in the nonconvex setting, which arising in many contemporary applications such as machine learning, statistics, and signal/image processing. To solve this problem, we propose a new nonmonotone accelerated proximal gradient method with variable stepsize strategy. Note that incorporating inertial term into proximal gradient method is a simple and efficient acceleration technique, while the descent property of the proximal gradient algorithm will lost. In our algorithm, the iterates generated by inertial proximal gradient scheme are accepted when the objective function values decrease or increase appropriately; otherwise, the iteration point is generated by proximal gradient scheme, which makes the function values on a subset of iterates are decreasing. We also introduce a variable stepsize strategy, which does not need a line search or does not need to know the Lipschitz constant and makes the algorithm easy to implement. We show that the sequence of iterates generated by the algorithm converges to a critical point of the objective function. Further, under the assumption that the objective function satisfies the Kurdyka–Łojasiewicz inequality, we prove the convergence rates of the objective function values and the iterates. Moreover, numerical results on both convex and nonconvex problems are reported to demonstrate the effectiveness and superiority of the proposed method and stepsize strategy.

Abstract

Nonnegative tensor factorizations (NTF) have applications in statistics, computer vision, exploratory multi-way data analysis, and blind source separation. This paper studies randomized multiplicative updating algorithms for symmetric NTF via random projections and random samplings. For random projections, we consider two methods to generate the random matrix and analyze the computational complexity, while for random samplings the uniform sampling strategy and its variants are examined. The mixing of these two strategies is then considered. Some theoretical results are presented based on the bounds of the singular values of sub-Gaussian matrices and the fact that randomly sampling rows from an orthogonal matrix results in a well-conditioned matrix. These algorithms are easy to implement, and their efficiency is verified via test tensors from both synthetic and real datasets, such as for clustering facial images.

It is possible to solve unbounded convex vector optimization problems (CVOPs) in two phases: (1) computing or approximating the recession cone of the upper image and (2) solving the equivalent bounded CVOP where the ordering cone is extended based on the first phase. In this paper, we consider unbounded CVOPs and propose an alternative solution methodology to compute or approximate the recession cone of the upper image. In particular, we relate the dual of the recession cone with the Lagrange dual of weighted sum scalarization problems whenever the dual problem can be written explicitly. Computing this set requires solving a convex (or polyhedral) projection problem. We show that this methodology can be applied to semidefinite, quadratic, and linear vector optimization problems and provide some numerical examples.