Search

This paper introduces cutting planes that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set S ∩ P , where S is a closed set, and P is a polyhedron. Given an oracle that provides the distance from a point to S, we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron. These cuts are generated from convex forbidden zones, or S-free sets, derived from the oracle. We also consider the special case of polynomial optimization. Accordingly we develop a theory of outer-product-free sets, where S is the set of real, symmetric matrices of the form x x T . All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify several families of such sets. These families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently. Computational experiments demonstrate the promise of our approach. [ABSTRACT FROM AUTHOR]

The growing use of optimization models to help decision making has created a demand for such tools that allow formulating and solving more models of real-world processes and systems related to human activity in which hypotheses are not verify in a way specific for classical optimization. One of the approaches for real-world extremum problems under uncertainty is interval-valued optimization. In this paper, a twice differentiable vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. In this paper, the first order necessary optimality conditions of Karush-Kuhn-Tucker type are proved for differentiable interval-valued vector optimization problems under the first order constraint qualification. If the interval-valued objective function is assumed to be twice weakly differentiable and constraints functions are assumed to be twice differentiable, then two types of second order necessary optimality conditions under two various constraint qualifications are proved for such smooth interval-valued vector optimization problems. Finally, in order to illustrate the Karush-Kuhn-Tucker type necessary optimality conditions established in the paper, an example of an interval-valued optimization is given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the split equality Minty variational inequality problem in reflexive real Banach spaces. Then we construct a single projection inertial algorithm for solving the introduced problem. We establish a strong convergence result with the assumption that the mappings under consideration are Lipschitz continuous and quasimonotone. We give some specific applications of the main result and finally provide a numerical example to demonstrate the workability of our method. [ABSTRACT FROM AUTHOR]

In this paper, a new appropriate diagonal matrix estimation of the Hessian is introduced by minimizing the Byrd and Nocedal function subject to the weak secant equation. The Hessian estimate is used to correct the framework of a nonmonotone trust region algorithm with the regularized quasi-Newton method. Moreover, to counteract the adverse effect of monotonicity, we introduce a new nonmonotone strategy. The global and superlinear convergence of the suggested algorithm is established under some standard conditions. The numerical experiments on unconstrained optimization test functions show that the new algorithm is efficient and robust. [ABSTRACT FROM AUTHOR]

Trust and reputation systems play a pivotal role in modern decentralized environments, fostering cooperation and mitigating risks in various online interactions. This paper introduces a variational formulation approach to model and analyze trust and reputation systems. By formulating trust and reputation as variational problems, this approach offers a novel perspective on understanding the underlying mechanisms governing trust establishment. The variational formulation provides a mathematical framework to determine the equilibrium weighted trust values, taking into account that each trustee tries to maximize its gain, namely the benefit minus the costs. The paper illustrates the applicability of this variational formulation through a pletora of simulations, demonstrating its effectiveness in modeling trust and reputation systems. Insights gained from this approach offer valuable guidance for the design and implementation of more reliable and efficient trust and reputation mechanisms in decentralized environments [ABSTRACT FROM AUTHOR]

In this paper, we undertake further investigation to alleviate the issue of limit cycling behavior in training generative adversarial networks (GANs) through the proposed predictive centripetal acceleration algorithm (PCAA). Specifically, we first derive the upper and lower complexity bounds of PCAA for a general bilinear game, with the last-iterate convergence rate notably improving upon previous results. Then, we combine PCAA with the adaptive moment estimation algorithm (Adam) to propose PCAA-Adam, for practical training of GANs to enhance their generalization capability. Finally, we validate the effectiveness of the proposed algorithm through experiments conducted on bilinear games, multivariate Gaussian distributions, and the CelebA dataset, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we initially enhance the Fletcher–Reeves (FR) conjugate parameter through a shrinkage multiplier, leading to a derivative-free two-term search direction and its extended spectral version. The results indicate that both search directions demonstrate sufficient descent and trust-region properties, irrespective of the line search method utilized. Then, by combining the hyperplane projection-based approach and inertia technique, we present two inertial FR-type conjugate gradient projection-based methods for solving constrained nonlinear equations. The global convergence of our methods is theoretically established, without requiring the monotonicity or pseudo-monotonicity of the underlying mapping, nor the Lipschitz continuity condition. Numerical experiments conducted on constrained nonlinear equations, as well as applications in regularized decentralized logistic regression problems and sparse signal restoration problems, have demonstrated the numerical efficacy of our methods. [ABSTRACT FROM AUTHOR]

In recent years, the sparse and group sparse optimization problem has attracted extensive attention due to its wide applications in statistics, bioinformatics, signal interpretation and machine learning, which yields the sparsity both in group-wise and element-wise. In this paper, the sparse and group sparse optimization problem with a nonsmooth loss function is considered, where the sparsity and group sparsity are induced by a penalty composed of a combination of ℓ 1 norm and ℓ 2 , 1 norm, so it is called the sparse group Lasso (SGLasso) problem. To solve this problem, the nonsmooth loss function is smoothed first. Then, based on the smooth approximation of the loss function, a smoothing composite proximal gradient (SCPG) algorithm is proposed. It is showed that any accumulation point of the sequence generated by SCPG algorithm is a global optimal solution of the problem. Moreover, it is proved that the convergence rate of the objective function value is O (1 k 1 - σ ) where σ ∈ (0.5 , 1) is a constant. Finally, numerical results illustrate that the proposed SCPG algorithm is effective and robust for sparse and group sparse optimization problems. Especially, compared with some popular algorithms, SCPG algorithm has obvious advantages in anti-outlier. [ABSTRACT FROM AUTHOR]

Decision making problems with ambiguous data often arise in numerous practical fields which can be formulated as optimization models in fuzzy environment. This paper develops a new mathematical model using a proposed methodology to efficiently solve a multi-objective linear fuzzy fractional programming problem in trapezoidal fuzzy environment and generate a set of nondominated solutions. The concept of fuzzy cuts with different degrees of satisfaction is implemented which transforms the fuzzy optimization into an equivalent interval valued optimization. Subsequently, interval valued linear functions approximate the fuzzy valued fractional functions based on Taylor's series expansion. Finally, a proposed concept using weighting sum approach with varying weight vectors is utilized to design a mathematical model which generates the set of nondominated solutions. Two numerical examples including an existing problem and an additional practical problem in the field of production, are solved for the illustration of the proposed model. The results of the numerical problems are comparatively discussed with graphical analysis to justify the feasibility and applicability of the proposed model. [ABSTRACT FROM AUTHOR]

The strategy presented in this paper differs significantly from existing approaches as we formulate the problem as a standard optimization problem of difference of convex functions. We have developed the necessary and sufficient conditions for global solutions in this standard form. The main challenge in the standard form arises from a constraint of the form g (t) ≥ 1 , where g is a convex function. We utilize the classical inequality between the weighted arithmetic and harmonic means to overcome this challenge. This enables us to express the optimality conditions as a convex geometric programming problem and employ a predictor-corrector primal-dual interior point method for its solution, with weights updated during the predictor phase. The interior point method solves the dual problem of geometric programming and obtains the primal solution through exponential transformation. We have implemented the algorithm in Fortran 90 and validated it using a set of test problems from the literature. The proposed method successfully solved all the test problems, and the computational results are presented alongside the tested problems and the corresponding solutions found. [ABSTRACT FROM AUTHOR]

Iterative hard thresholding (IHT) and compressive sampling matching pursuit (CoSaMP) are two mainstream compressed sensing algorithms using the hard thresholding operator. The guaranteed performance of the two algorithms for signal recovery was mainly analyzed in terms of the restricted isometry property (RIP) of sensing matrices. At present, the best known bound using the RIP of order 3k for guaranteed performance of IHT (with the unit stepsize) is δ 3 k < 1 / 3 ≈ 0.5774 , and the bound for CoSaMP using the RIP of order 4k is δ4k < 0.4782. A fundamental question in this area is whether such theoretical results can be further improved. The purpose of this paper is to affirmatively answer this question and to rigorously show that the above-mentioned RIP bound for guaranteed performance of IHT can be significantly improved to δ 3 k < (5 − 1) / 2 ≈ 0.618 , and the bound for CoSaMP can be improved to δ4k < 0.5102. [ABSTRACT FROM AUTHOR]

A previous authors' paper introduces an accelerated composite gradient (ACG) variant, namely AC-ACG, for solving nonconvex smooth composite optimization (N-SCO) problems. In contrast to other ACG variants, AC-ACG estimates the local upper curvature of the N-SCO problem by using the average of the observed upper-Lipschitz curvatures obtained during the previous iterations, and uses this estimation and two composite resolvent evaluations to compute the next iterate. This paper presents an alternative FISTA-type ACG variant, namely AC-FISTA, which has the following additional features: (i) it performs an average of one composite resolvent evaluation per iteration; and (ii) it estimates the local upper curvature by using the average of the previously observed upper (instead of upper-Lipschitz) curvatures. These two properties acting together yield a practical AC-FISTA variant which substantially outperforms earlier ACG variants, including the AC-ACG variants discussed in the aforementioned authors' paper. [ABSTRACT FROM AUTHOR]