An approximate Hahn–Banach theorem for positively homogeneous functions

This note provides an approximate version of the Hahn–Banach theorem for non-necessarily convex extended-real valued positively homogeneous functions of degree one. Given p : X → R∪{+∞} such a function defined on the real vector space X, and a linear function defined on a subspace V of X and dominated by p (i.e. (x) ≤ p(x) for all x ∈ V), we say that can approximately be p-extended to X, if is the pointwise limit of a net of linear functions on V, every one of which can be extended to a linear function defined on X and dominated by p. The main result of this note proves that can approximately be p-extended to X if and only if is dominated by p∗∗, the pointwise supremum over the family of all the linear functions on X which are dominated by p.

Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem

In recent times the Douglas–Rachford algorithm has been observed empirically to solve a variety of nonconvex feasibility problems including those of a combinatorial nature. For many of these problems current theory is not sufficient to explain this observed success and is mainly concerned with questions of local convergence. In this paper we analyze global behavior of the method for finding a point in the intersection of a half-space and a potentially non-convex set which is assumed to satisfy a well-quasi-ordering property or a property weaker than compactness. In particular, the special case in which the second set is finite is covered by our framework and provides a prototypical setting for combinatorial optimization problems.

Stability of Supporting and Exposing Elements of Convex Sets in Banach Spaces

* This work was supported by the CNR while the author was visiting the University of Milan.

Exact Discriminant Function Design Using Some Optimization Techniques

Some aspects of design of the discriminant functions that in the best way separate points of predefined final sets are considered. The concept is introduced of the nested discriminant functions which allow to separate correctly points of any of the final sets. It is proposed to apply some methods of non-smooth optimization to solve arising extremal problems efficiently.

Solution of the Least Squares Method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima.

The kernel is in the least core for permutation games

Permutation games are totally balanced transferable utility cooperative games arising from certain sequencing and re-assignment optimization problems. It is known that for permutation games the bargaining set and the core coincide, consequently, the kernel is a subset of the core. We prove that for permutation games the kernel is contained in the least core, even if the latter is a lower dimensional subset of the core. By means of a 5-player permutation game we demonstrate that, in sense of the lexicographic center procedure leading to the nucleolus, this inclusion result can not be strengthened. Our 5-player permutation game is also an example (of minimum size) for a game with a non-convex kernel.

The Development of a Hybrid Optimization Algorithm for the Evaluation and Optimization of the Asynchronous Pulse Unit

The effectiveness of an optimization algorithm can be reduced to its ability to navigate an objective function’s topology. Hybrid optimization algorithms combine various optimization algorithms using a single meta-heuristic so that the hybrid algorithm is more robust, computationally efficient, and/or accurate than the individual algorithms it is made of. This thesis proposes a novel meta-heuristic that uses search vectors to select the constituent algorithm that is appropriate for a given objective function. The hybrid is shown to perform competitively against several existing hybrid and non-hybrid optimization algorithms over a set of three hundred test cases. This thesis also proposes a general framework for evaluating the effectiveness of hybrid optimization algorithms. Finally, this thesis presents an improved Method of Characteristics Code with novel boundary conditions, which better characterizes pipelines than previous codes. This code is coupled with the hybrid optimization algorithm in order to optimize the operation of real-world piston pumps.

Intertemporal and spatial location of disposal facilities

The optimal capacities and locations of a sequence of landfills are studied, and the interactions between these characteristics are considered. Deciding the capacity of a landfill has some spatial implications since it affects the feasible region for the remaining landfills, and some temporal implications because the capacity determines the lifetime of the landfill and hence the moment of time when the next landfills should be constructed. Some general mathematical properties of the solution are provided and interpreted from an economic point of view. The resulting problem turns out to be non-convex and, therefore, it cannot be solved by conventional optimization techniques. Some global optimization methods are used to solve the problem in a particular case in order to illustrate how the solution depends on the parameter values.

JOINT ENERGY BEAMFORMING AND TIME ASSIGNMENT FOR COOPERATIVE WIRELESS POWERED NETWORKS

Wireless power transfer (WPT) and radio frequency (RF)-based energy har- vesting arouses a new wireless network paradigm termed as wireless powered com- munication network (WPCN), where some energy-constrained nodes are enabled to harvest energy from the RF signals transferred by other energy-sufficient nodes to support the communication operations in the network, which brings a promising approach for future energy-constrained wireless network design. In this paper, we focus on the optimal WPCN design. We consider a net- work composed of two communication groups, where the first group has sufficient power supply but no available bandwidth, and the second group has licensed band- width but very limited power to perform required information transmission. For such a system, we introduce the power and bandwidth cooperation between the two groups so that both group can accomplish their expected information delivering tasks. Multiple antennas are employed at the hybrid access point (H-AP) to en- hance both energy and information transfer efficiency and the cooperative relaying is employed to help the power-limited group to enhance its information transmission throughput. Compared with existing works, cooperative relaying, time assignment, power allocation, and energy beamforming are jointly designed in a single system. Firstly, we propose a cooperative transmission protocol for the considered system, where group 1 transmits some power to group 2 to help group 2 with information transmission and then group 2 gives some bandwidth to group 1 in return. Sec- ondly, to explore the information transmission performance limit of the system, we formulate two optimization problems to maximize the system weighted sum rate by jointly optimizing the time assignment, power allocation, and energy beamforming under two different power constraints, i.e., the fixed power constraint and the aver- age power constraint, respectively. In order to make the cooperation between the two groups meaningful and guarantee the quality of service (QoS) requirements of both groups, the minimal required data rates of the two groups are considered as constraints for the optimal system design. As both problems are non-convex and have no known solutions, we solve it by using proper variable substitutions and the semi-definite relaxation (SDR). We theoretically prove that our proposed solution method can guarantee to find the global optimal solution. Thirdly, consider that the WPCN has promising application potentials in future energy-constrained net- works, e.g., wireless sensor network (WSN), wireless body area network (WBAN) and Internet of Things (IoT), where the power consumption is very critical. We investigate the minimal power consumption optimal design for the considered co- operation WPCN. For this, we formulate an optimization problem to minimize the total consumed power by jointly optimizing the time assignment, power allocation, and energy beamforming under required data rate constraints. As the problem is also non-convex and has no known solutions, we solve it by using some variable substitutions and the SDR method. We also theoretically prove that our proposed solution method for the minimal power consumption design guarantees the global optimal solution. Extensive experimental results are provided to discuss the system performance behaviors, which provide some useful insights for future WPCN design. It shows that the average power constrained system achieves higher weighted sum rate than the fixed power constrained system. Besides, it also shows that in such a WPCN, relay should be placed closer to the multi-antenna H-AP to achieve higher weighted sum rate and consume lower total power.

Efficient and precise interactive hand tracking through joint, continuous optimization of pose and correspondences

Fully articulated hand tracking promises to enable fundamentally new interactions with virtual and augmented worlds, but the limited accuracy and efficiency of current systems has prevented widespread adoption. Today's dominant paradigm uses machine learning for initialization and recovery followed by iterative model-fitting optimization to achieve a detailed pose fit. We follow this paradigm, but make several changes to the model-fitting, namely using: (1) a more discriminative objective function; (2) a smooth-surface model that provides gradients for non-linear optimization; and (3) joint optimization over both the model pose and the correspondences between observed data points and the model surface. While each of these changes may actually increase the cost per fitting iteration, we find a compensating decrease in the number of iterations. Further, the wide basin of convergence means that fewer starting points are needed for successful model fitting. Our system runs in real-time on CPU only, which frees up the commonly over-burdened GPU for experience designers. The hand tracker is efficient enough to run on low-power devices such as tablets. We can track up to several meters from the camera to provide a large working volume for interaction, even using the noisy data from current-generation depth cameras. Quantitative assessments on standard datasets show that the new approach exceeds the state of the art in accuracy. Qualitative results take the form of live recordings of a range of interactive experiences enabled by this new approach.

An algorithm for computerized automatic tuning of power system stabilizers

The main objective of this paper is to relieve the power system engineers from the burden of the complex and time-consuming process of power system stabilizer (PSS) tuning. To achieve this goal, the paper proposes an automatic process for computerized tuning of PSSs, which is based on an iterative process that uses a linear matrix inequality (LMI) solver to find the PSS parameters. It is shown in the paper that PSS tuning can be written as a search problem over a non-convex feasible set. The proposed algorithm solves this feasibility problem using an iterative LMI approach and a suitable initial condition, corresponding to a PSS designed for nominal operating conditions only (which is a quite simple task, since the required phase compensation is uniquely defined). Some knowledge about the PSS tuning is also incorporated in the algorithm through the specification of bounds defining the allowable PSS parameters. The application of the proposed algorithm to a benchmark test system and the nonlinear simulation of the resulting closed-loop models demonstrate the efficiency of this algorithm. (C) 2009 Elsevier Ltd. All rights reserved.

Multiobjective Particle Swarm Approach for the Design of a Brushless DC Wheel Motor

The roots of swarm intelligence are deeply embedded in the biological study of self-organized behaviors in social insects. Particle swarm optimization (PSO) is one of the modern metaheuristics of swarm intelligence, which can be effectively used to solve nonlinear and non-continuous optimization problems. The basic principle of PSO algorithm is formed on the assumption that potential solutions (particles) will be flown through hyperspace with acceleration towards more optimum solutions. Each particle adjusts its flying according to the flying experiences of both itself and its companions using equations of position and velocity. During the process, the coordinates in hyperspace associated with its previous best fitness solution and the overall best value attained so far by other particles within the group are kept track and recorded in the memory. In recent years, PSO approaches have been successfully implemented to different problem domains with multiple objectives. In this paper, a multiobjective PSO approach, based on concepts of Pareto optimality, dominance, archiving external with elite particles and truncated Cauchy distribution, is proposed and applied in the design with the constraints presence of a brushless DC (Direct Current) wheel motor. Promising results in terms of convergence and spacing performance metrics indicate that the proposed multiobjective PSO scheme is capable of producing good solutions.

Compact clinical MRI magnet design using a multilayer current density approach

In this work, a new method of optimization is successfully applied to the theoretical design of compact, actively shielded, clinical MRI magnets. The problem is formulated as a two-step process in which the desired current densities on multiple, cc-axial surface layers are first calculated by solving Fredholm equations of the first kind. Non-linear optimization methods with inequality constraints are then invoked to fit practical magnet coils to the desired current densities. The current density approach allows rapid prototyping of unusual magnet designs. The emphasis of this work is on the optimal design of short, actively-shielded MRI magnets for whole-body imaging. Details of the hybrid numerical model are presented, and the model is used to investigate compact, symmetric, and asymmetric MRI magnets. Magnet designs are presented for actively-shielded, symmetric magnets of coil length 1.0 m, which is considerably shorter than currently available designs of comparable dsv size. Novel, actively-shielded, asymmetric magnet designs are also presented in which the beginning of a 50-cm dsv is positioned just 11 cm from the end of the coil structure, allowing much improved access to the patient and reduced patient claustrophobia. Magn Reson Med 45:331540, 2001. (C) 2001 Wiley-Liss, Inc.

Direct-search penalty/barrier methods

In Nonlinear Optimization Penalty and Barrier Methods are normally used to solve Constrained Problems. There are several Penalty/Barrier Methods and they are used in several areas from Engineering to Economy, through Biology, Chemistry, Physics among others. In these areas it often appears Optimization Problems in which the involved functions (objective and constraints) are non-smooth and/or their derivatives are not know. In this work some Penalty/Barrier functions are tested and compared, using in the internal process, Derivative-free, namely Direct Search, methods. This work is a part of a bigger project involving the development of an Application Programming Interface, that implements several Optimization Methods, to be used in applications that need to solve constrained and/or unconstrained Nonlinear Optimization Problems. Besides the use of it in applied mathematics research it is also to be used in engineering software packages.

Manipulator trajectory planning using a MOEA

Generating manipulator trajectories considering multiple objectives and obstacle avoidance is a non-trivial optimization problem. In this paper a multi-objective genetic algorithm based technique is proposed to address this problem. Multiple criteria are optimized considering up to five simultaneous objectives. Simulation results are presented for robots with two and three degrees of freedom, considering two and five objectives optimization. A subsequent analysis of the spread and solutions distribution along the converged non-dominated Pareto front is carried out, in terms of the achieved diversity.