On the acceleration of the double smoothing technique for unconstrained convex optimization problems

In this paper, energy efficient power control for small cells underlaying a macro cellular network is investigated. We formulate the power control problem in self-organizing small cell networks as a non-cooperative game, and propose a distributed energy efficient power control scheme, which allows the small base stations (SBSs) to take individual decisions for attaining the Nash equilibrium (NE) with minimum information exchange. Specially, in the non-cooperative power control game, a non-convex optimization problem is formulated for each SBS to maximize their energy efficiency (EE). By exploiting the properties of parameter-free fractional programming and the concept of perspective function, the non-convex optimization problem for each SBS is transformed into an equivalent constrained convex optimization problem. Then, the constrained convex optimization problem is converted into an unconstrained convex optimization problem by exploiting the mixed penalty function method. The inequality constraints are eliminated by introducing the logarithmic barrier functions and the equality constraint is eliminated by introducing the quadratic penalty function. We also theoretically show the existence and the uniqueness of the NE in the non-cooperative power control game. Simulation results show remarkable improvements in terms of EE by using the proposed scheme.

This paper gives a new subspace correction algorithm for nonlinear unconstrained convex optimization problems based on the multigrid approach proposed by S. Nash in 2000 and the subspace correction algorithm proposed by X. Tai and J. Xu in 2001. Under some reasonable assumptions, we obtain the convergence as well as a convergence rate estimate for the algorithm. Numerical results show that the algorithm is effective.

A modified Polak–Ribière–Polyak conjugate gradient algorithm for nonsmooth convex programs

In this work we study the problem of unconstrained convex- optimization in a fully distributed multi-agent setting, which includes asynchronous computation and lossy communication. In particular, we extend a recently proposed algorithm named Newton-Raphson Consensus by integrating it with a broadcast-based average consensus algorithm which is robust to packet losses. We show via the separation of time scales principle that under mild conditions (i.e., persistency of the agents activation and bounded consecutive communication failures) the proposed algorithm is proved to be locally exponentially stable with respect to the optimal global solution. Finally, we complement the theoretical analysis with numerical simulations and comparisons based on real datasets.

Matching individuals within a group of spatially nonoverlapping surveillance cameras, also known as person reidentification, has recently attracted a lot of research interest. Current methods mainly focus on feature representation or distance measure, which directly compare person images captured by different cameras. However, it is still a problem because of various surveillance conditions; for example, view switching, lighting variations, and image scaling. Although the brightness transfer function was proposed to address the problem of illumination variation, it could not handle view and scale changes among various cameras. In this paper, we propose a new approach to compensate for the inconsistency of feature distributions of person images captured by different cameras. More precisely, a feature projection matrix (FPM) is learned to project image features of one camera to the feature space of another camera, from which the latent device difference can be effectively eliminated for the person reidentification task. In particular, we formulate the FPM learning as a smooth unconstrained convex optimization problem and use a simple gradient descent algorithm with stochastic samples to accelerate the solving process. Extensive comparative experiments conducted on three standard datasets have shown the promising prospect of the proposed method.

We study networked unconstrained convex optimization problems where the objective function changes continuously in time. We propose a decentralized algorithm (DePCoT) with a discrete time-sampling scheme to find and track the solution trajectory based on prediction and gradient-based correction steps, while sampling the problem data at a constant sampling period h. Under suitable conditions and for limited sampling periods, we establish that the asymptotic error bound behaves as O(h2), which outperforms the state of the art existing error bound of O(h) for correction-only methods. The key contributions are the prediction step and a decentralized method to approximate the inverse of the Hessian of the cost function in a decentralized way, which yields quantifiable trade-offs between communication and accuracy.

We devise a novel quasi-Newton algorithm for solving unconstrained convex optimization problems. The proposed algorithm is built on our previous framework of the iteratively preconditioned gradient-descent (IPG) algorithm. IPG utilized Richardson iteration to update a preconditioner matrix that approximates the inverse of the Hessian matrix. In this letter, we substitute the Richardson iteration with a successive over-relaxation (SOR) formulation. The convergence guarantee of the proposed algorithm and its theoretical improvement over vanilla IPG are presented. The algorithm is used in a mobile robot position estimation problem for numerical validation using a moving horizon estimation (MHE) formulation. Compared with IPG, the results demonstrate an improved performance of the proposed algorithm in terms of computational time and the number of iterations needed for convergence, matching our theoretical results.

A warm-start FE-dABCD algorithm for elliptic optimal control problems with constraints on the control and the gradient of the state

Inertial primal-dual projection neurodynamic approaches for constrained convex optimization problems and application to sparse recovery.

The standard assumption for proving linear convergence of first order methods for smooth convex optimization is the strong convexity of the objective function, an assumption which does not hold for many practical applications. In this paper, we derive linear convergence rates of several first order methods for solving smooth non-strongly convex constrained optimization problems, i.e. involving an objective function with a Lipschitz continuous gradient that satisfies some relaxed strong convexity condition. In particular, in the case of smooth constrained convex optimization, we provide several relaxations of the strong convexity conditions and prove that they are sufficient for getting linear convergence for several first order methods such as projected gradient, fast gradient and feasible descent methods. We also provide examples of functional classes that satisfy our proposed relaxations of strong convexity conditions. Finally, we show that the proposed relaxed strong convexity conditions cover important applications ranging from solving linear systems, Linear Programming, and dual formulations of linearly constrained convex problems.

In this article, we propose a new line search method for solving unconstrained optimization problems in that we combine a nonmonotone strategy into a modified Armijo rule and design a new algorithm that possibly chooses a larger steplength. This can decrease the number of iterations and function evaluations and can improve the efficiency of the algorithm. The global convergence and convergence rate are analysed under some suitable conditions. Preliminary numerical experiments establish that the new approach is robust and efficient for unconstrained optimization problems.

Stochastic traffic assignment, Lagrangian dual, and unconstrained convex optimization

The application of Goeken–Johnson’s Runge–Kutta methods in unconstrained convex optimization

A popular optimization technique called the conjugate gradient method (CGM) is renowned for its effectiveness in addressing problems involving unconstrained optimization. Several conjugate gradient (CG) techniques have been proven to possess global convergence properties both theoretically and numerically. The Dai-Liao-type CGM is a variant that incorporates certain modifications to enhance its convergence properties. This paper examines the global convergence properties of a Dai-Liao-type CGM for unconstrained optimization problems. Theoretically, this study investigates the conditions under which the method ensures convergence to the global minimum of the objective function, focusing on the algorithm’s descent directions, the necessary reduction in objective function values, and termination criteria. A numerical experiment is conducted on a set of unconstrained optimization problems to validate the theoretical results obtained in this work. The numerical findings of this study demonstrate the robustness and reliability of the Dai-Liao-type CGM, showing its ability to find the global optimal solution in a wide range of unconstrained optimization problems.

A parameter-free optimization technique is applied in Quasi-Newton’s method for solving unconstrained multiobjective optimization problems. The components of the Hessian matrix are constructed using q-derivative, which is positive definite at every iteration. The step-length is computed by an Armijo-like rule which is responsible to escape the point from local minimum to global minimum at every iteration due to q-derivative. Further, the rate of convergence is proved as a superlinear in a local neighborhood of a minimum point based on q-derivative. Finally, the numerical experiments show better performance.