Exact Optimal Value Research Articles

Tight Sum-of-squares Lower Bounds for Binary Polynomial Optimization Problems

For binary polynomial optimization problems of degree 2d with n variables Sakaue, Takeda, Kim, and Ito [ 33 ] proved that the \(\lceil \frac{n+2d-1}{2}\rceil\) th semidefinite (SDP) relaxation in the SoS/Lasserre hierarchy of SDP relaxations provides the exact optimal value. When n is an odd number, we show that their analysis is tight, i.e., we prove that \(\frac{n+2d-1}{2}\) levels of the SoS/Lasserre hierarchy are also necessary. Laurent [ 24 ] showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull of a linear representation of a set with no integral points. She conjectured that the SoS/Lasserre rank for the same problem is n -1. In this article, we disprove this conjecture and derive lower and upper bounds for the rank.

Optimal Softening for Gravitational Force Calculations in N-body Dynamics

Abstract The choice of the optimal value of the softening length (ϵ i ) of each particle dealing with N-body simulations has a profound impact on error values in the gravitational force calculation. A slight deviation from its exact optimal value may result in a large error in the calculation. In this paper we augment the accuracy of the existing gravitational force calculation methods by providing a new technique to calculate the individual optimal values of ϵ i for various configurations of the Plummer density model. We have proposed an expression that relates each particle by considering the average characteristic length (λ i ) and density (ρ i ), unlike previous studies that considered ϵ i as an exclusive function of ρ i . We have performed gravitational force calculations for each and every particle from the Plummer density model using compact as well as noncompact gravitational force methods based on smoothed particle hydrodynamics. We have tested our new equation for the entire range of numerical simulations performed during the study. It is found that the errors in our force calculations are not only lower than those estimated from previous studies but also remain flat for various considerations of nearest neighboring particles (N neigh). The adjusted expression of ϵ λ,i in our study has less dependence on N neigh in the Plummer sphere. Finally, based on the results obtained using the method proposed in this study, we find that it remarkably improves both the accuracy as well as the stability of the gravitational force calculation.

Robust Vision-Based Tube Model Predictive Control of Multiple Mobile Robots for Leader–Follower Formation

Generally, vision-based controls use various camera sensors and require camera calibration, while the control performance would degrade due to inaccuracy calibration. Therefore, in this paper, the proposed controller only makes use of the image information from an un-calibrated perspective camera mounted at the follower robot without relative position measurement or any communication among the robots. First, the nominal visual formation kinematic model is developed using the camera models. Then it is redescribed as a quadratic programming (QP) with the specified constraints. A neurodynamic optimization based on primal-dual neural network is utilized to ensure the QP being converged to the exact optimal values. Through two-time-scale neuro-dynamical optimization, the gain scheduling of the ancillary state feedback can be realized so that the state variables are constrained within an invariant designed tube. The experiment results provide the verification for the effectiveness of the proposed approach.

Quantized Gradient-Descent Algorithm for Distributed Resource Allocation

In this paper, we consider a distributed resource allocation problem with communication limitation. We propose a gradient-descent algorithm to solve the distributed resource allocation problem with quantization mechanism due to the communication limitations or in order to reduce the communication cost in the network. With carefully selected parameters, the convergence and correctness of the quantized algorithm can be obtained for fixed communication topologies and moreover, extended to switching jointly connected topologies. The exact optimal value can be obtained, which is different from some existing works, whose convergence accuracies were limited by the quantization bandwidth. In addition, the data rate of the proposed algorithm is also analyzed for both fixed and switching communication networks. Particularly, in the fixed topology case, we obtain that 1 bit is sufficient to solve the distributed problem with the proposed algorithm.

The optimal noise-rejection threshold for normal and impaired hearing

Binary masking represents a powerful tool for increasing speech intelligibility in noise. An essential aspect involves the local criterion (LC), which defines the signal-to-noise ratio below which time-frequency units are discarded. But binary masking is a victim of its own success in one regard—it produces ceiling sentence intelligibility across a broad range of LC values, making the exact optimal LC value difficult to determine. Further, the optimal value for hearing-impaired (HI) listeners is largely unknown. In the current study, the optimal LC was determined in normal-hearing (NH) and HI listeners using speech materials less likely to produce ceiling effects. The CID W22 words were mixed with noise consisting of recordings from a busy hospital cafeteria, then subjected to ideal binary masking. LC values ranged from -20 to + 5 dB relative to the overall SNR of -8 dB. NH subjects were tested at 65 dBA and HI subjects were tested at 65 dBA plus NAL-RP hearing-aid gains. Preliminary results suggest that the optimal LC is similar for NH and HI listeners. Additional conditions involving different speech materials and noise types suggest that the optimal LC can vary as a function of speech and/or noise type. [Work supported by NIH.]

Optimal hematocrit in an artificial microvascular network.

Higher hematocrit increases the oxygen-carrying capacity of blood but also increases blood viscosity, thus decreasing blood flow through the microvasculature and reducing the oxygen delivery to tissues. Therefore, an optimal value of hematocrit that maximizes tissue oxygenation must exist. We used viscometry and an artificial microvascular network device to determine the optimal hematocrit in vitro. Suspensions of fresh red blood cells (RBCs) in plasma, normal saline, or a protein-containing buffer and suspensions of stored red blood cells (at Week 6 of standard hypothermic storage) in plasma with hematocrits ranging from 10 to 80% were evaluated. For viscometry, optimal hematocrits were 10, 25.2, 31.9, 37.1, and 37.5% for fresh RBCs in plasma at shear rates of 3.2 or less, 11.0, 27.7, 69.5, and 128.5 inverse seconds. For the artificial microvascular network, optimal hematocrits were 51.1, 55.6, 59.2, 60.9, 62.3, and 64.6% for fresh RBCs in plasma and 46.4, 48.1, 54.8, 61.4, 65.7, and 66.5% for stored RBCs in plasma at pressures of 2.5, 5, 10, 20, 40, and 60 cm H2 O. Although exact optimal hematocrit values may depend on specific microvascular architecture, our results suggest that the optimal hematocrit for oxygen delivery in the microvasculature depends on perfusion pressure. Therefore, anemia in chronic disorders may represent a beneficial physiological response to reduced perfusion pressure resulting from decreased heart function and/or vascular stenosis. Our results may help explain why a therapeutically increasing hematocrit in such conditions with RBC transfusion frequently leads to worse clinical outcomes.

Visual Servoing of Constrained Mobile Robots Based on Model Predictive Control

This paper develops an image-based visual servoing (IBVS) control strategy using model predictive control (MPC) to stabilize a physically constrained mobile robot. In IBVS strategy, ambiguity, and degeneracy problems of the homography and fundamental matrix-based algorithms can be avoided. Moreover, a synthetic error vector incorporating the advantages of IBVS and position-based visual servoing is defined that includes both the robot angle and image coordinates. By using linear system control theory, the kinematics of nonholonomic chained robotic systems can be transformed into a skew-symmetric form, and through introducing an exponential decay phase, the uncontrollable problem can be solved. Then, an MPC strategy is developed and, thereafter, iteratively transformed into a constrained quadratic programming (QP) problem. Subsequently, we utilize a primal-dual neural network (PDNN) to solve this QP problem. By using PDNN optimization, the cost function of MPC effectively converges to the exact optimal values. Finally, experimental studies on the actual robotic systems have been conducted to demonstrate the performance of the proposed approach.

Exact Semidefinite Programming Relaxations with Truncated Moment Matrix for Binary Polynomial Optimization Problems

For binary polynomial optimization problems (POPs) of degree $d$ with $n$ variables, we prove that the $\lceil(n+d-1)/2\rceil$th semidefinite programming (SDP) relaxation in Lasserre's hierarchy of SDP relaxations provides the exact optimal value. If binary POPs involve only even-degree monomials, we show that it can be further reduced to $\lceil(n+d-2)/2\rceil$. This bound on the relaxation order coincides with the conjecture by Laurent in 2003, which was recently proved by Fawzi, Saunderson, and Parrilo, on binary quadratic optimization problems where $d=2$. We also numerically confirm that the bound is tight. More precisely, we present instances of binary POPs that require solving at least the $\lceil(n+d-1)/2\rceil$th SDP relaxation for general binary POPs and the $\lceil(n+d-2)/2\rceil$th SDP relaxation for even-degree binary POPs to obtain the exact optimal values.

Trajectory-Tracking Control of Mobile Robot Systems Incorporating Neural-Dynamic Optimized Model Predictive Approach

Mobile robots tracking a reference trajectory are constrained by the motion limits of their actuators, which impose the requirement for high autonomy driving capabilities in robots. This paper presents a model predictive control (MPC) scheme incorporating neural-dynamic optimization to achieve trajectory tracking of nonholonomic mobile robots (NMRs). By using the derived tracking-error kinematics of nonholonomic robots, the proposed MPC approach is iteratively transformed as a constrained quadratic programming (QP) problem, and then a primal–dual neural network is used to solve this QP problem over a finite receding horizon. The applied neural-dynamic optimization can make the cost function of MPC converge to the exact optimal values of the formulated constrained QP. Compared with the existing fast MPC, which requires repeatedly calculating the Hessian matrix of the Langragian and then solves a quadratic program. The computation complexity reaches ${O}({n}^{{3}})$ , while the proposed neural-dynamic optimization contains ${O}({n}^{{2}})$ operations. Finally, extensive experiments are provided to illustrate that the MPC scheme has an effective performance on a real mobile robot system.

Robustness of optimal random searches in fragmented environments.

The random search problem is a challenging and interdisciplinary topic of research in statistical physics. Realistic searches usually take place in nonuniform heterogeneous distributions of targets, e.g., patchy environments and fragmented habitats in ecological systems. Here we present a comprehensive numerical study of search efficiency in arbitrarily fragmented landscapes with unlimited visits to targets that can only be found within patches. We assume a random walker selecting uniformly distributed turning angles and step lengths from an inverse power-law tailed distribution with exponent μ. Our main finding is that for a large class of fragmented environments the optimal strategy corresponds approximately to the same value μ(opt)≈2. Moreover, this exponent is indistinguishable from the well-known exact optimal value μ(opt)=2 for the low-density limit of homogeneously distributed revisitable targets. Surprisingly, the best search strategies do not depend (or depend only weakly) on the specific details of the fragmentation. Finally, we discuss the mechanisms behind this observed robustness and comment on the relevance of our results to both the random search theory in general, as well as specifically to the foraging problem in the biological context.

Comments on “Optimal Use of Mixed Catalysts for Two Successive Chemical Reactions”

The paper "Optimal use of mixed catalysts for two successive chemical reactions" by Jackson was first presenting an analytical solution for the plug flow reactor catalyst blend problem. However, the exact optimal value of the problem was not given, which has caused confusion and inaccuracies in the subsequent literatures. As a supplement and correction, this study gives the optimal solutions and optimal objective values analytically for the cases when the reactor length equals 1 and 12. These results are also verified numerically using the time-scaling method.

Quadratic third-order tensor optimization problem with quadratic constraints

Quadratically constrained quadratic programs (QQPs) problems play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Semidenite programming (SDP) relaxations often provide good approximate solutions to these hard problems. For several special cases of QQP, e.g., convex programs and trust region subproblems, SDP relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective.In this paper, we consider a certain QQP where the variable is neither vector nor matrix but a third-order tensor. This problem can be viewed as a generalization of the ordinary QQP with vector or matrix as it's variant. Under some mild conditions, we rst show that SDP relaxation provides exact optimal solutions for the original problem. Then we focus on two classes of homogeneous quadratic tensor programming problems which have no requirements on the constraints number. For one, we provide an easily implemental polynomial time algorithm to approximately solve the problem and discuss the approximation ratio. For the other, we show there is no gap between the SDP relaxation and itself.

Quadratic third-order tensor optimization problem with quadratic constraints

Quadratically constrained quadratic programs (QQPs) problems play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Semidenite programming (SDP) relaxations often provide good approximate solutions to these hard problems. For several special cases of QQP, e.g., convex programs and trust region subproblems, SDP relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective.In this paper, we consider a certain QQP where the variable is neither vector nor matrix but a third-order tensor. This problem can be viewed as a generalization of the ordinary QQP with vector or matrix as it's variant. Under some mild conditions, we rst show that SDP relaxation provides exact optimal solutions for the original problem. Then we focus on two classes of homogeneous quadratic tensor programming problems which have no requirements on the constraints number. For one, we provide an easily implemental polynomial time algorithm to approximately solve the problem and discuss the approximation ratio. For the other, we show there is no gap between the SDP relaxation and itself.

Hierarchical collapsing method for direct defuzzification of general type-2 fuzzy sets

Karnik–Mendel (KM) and Enhanced KM (EKM) algorithms are proved to be efficient approaches when dealing with general type-2 fuzzy sets represented by their α-planes. On the other hand, collapsing method, as a type-reduction algorithm for interval type-2 fuzzy sets has shown to be an effectual type-reduction method for interval type-2 fuzzy sets. In this paper, we aim at developing mathematical formulations based on collapsing procedure and α-plane decomposition in order to utilize geometric properties of general type-2 fuzzy sets for preparing a direct defuzzification procedure. The proposed methods are fast and robust. Extensive simulations have shown that the proposed algorithm outperforms most of the existing approaches in this field and its results are closer to the exact optimal defuzzified values.

A new predictor–corrector scheme for valuing American puts

In this paper, we present a new numerical scheme, based on the finite difference method, to solve American put option pricing problems. Upon applying a Landau transform or the so-called front-fixing technique [19] to the Black–Scholes partial differential equation, a predictor–corrector finite difference scheme is proposed to numerically solve the nonlinear differential system. Through the comparison with Zhu’s analytical solution [35], we shall demonstrate that the numerical results obtained from the new scheme converge well to the exact optimal exercise boundary and option values. The results of our numerical examples suggest that this approach can be used as an accurate and efficient method even for pricing other types of financial derivative with American-style exercise.

Optimum tuning of series and parallel LR circuits for passive vibration suppression using piezoelectric elements

Bending vibration of flexible structures can be suppressed passively using piezoelectric electromechanical transducers and optimally tuned LR circuits. Since these systems include both mechanical and electrical elements, the governing equations consist of electrically coupled equations of motion. This paper describes a new method for deriving the governing equations that describe a system’s vibration suppression based on the equilibrium of force principle and using an equivalent mechanical model of a piezoelectric element. Both series and parallel LR circuits are considered in the modeling approach. The optimum values for a mechanical vibration absorber can be formulated by using the two fixed points method. However, exact optimal values for the resistances of the LR circuits have not been formulated in the research literature thus far, and approximate values have been used. Analytical formulations are derived in this paper, and optimum values of the LR circuits are presented, not only in displacement, but also in terms of velocity and acceleration. The effects of the stiffness of the adhesive bond between the host structure and piezoelectric element, the dielectric loss in a piezoelectric element, and the internal resistance of an inductor are considered in the theoretical analysis. The effectiveness of the described analytical method is validated through simulations and experiments.

Complement to a comparative analysis of heuristics for the p-median problem

A recent comparison of evolutionary, neural network, and scatter search heuristics for solving the p-median problem is completed by (i) gathering or obtaining exact optimal values in order to evaluate errors precisely, and (ii) including results obtained with several variants of a variable neighborhood search (VNS) heuristic. For a first, well-known, series of instances, the average errors of the evolutionary and neural network heuristics are over 10% and more than 1000 times larger than that of VNS. For a second series, this error is about 3% while the errors of the parallel VNS and of a hybrid heuristic are about 0.01% and that of parallel scatter search even smaller.

The final N ETLIB-LP results

With standard linear programming solvers there is always some uncertainty about the precise values of the optimal solutions. We implemented a program using exact rational arithmetic to compute proofs for the feasibility and optimality of an LP solution. This paper reports the exact optimal objective values for all N ETLIB problems.

Inexact genetic algorithm approach to target values setting of engineering requirements in QFD

In Quality Function Deployment, determination of the target values for engineering requirements is the most difficult process. A fuzzy optimization model is presented for the determination of target values for engineering requirements in Quality Function Deployment. An inexact genetic algorithm approach was introduced to solve the model that takes the mutation along the weighted gradient direction as a genetic operator. Instead of obtaining one set of exact optimal target values, the approach can generate a family of inexact optimal target values setting within an acceptable satisfaction degree. Through an interactive approach, a design team can determine a combination of preferred solution sets from which a set of preferred target values of engineering requirements based on a specific design scenario can be obtained. An example of car door design is used to illustrate the approach.

A note on lack of strong duality for quadratic problems with orthogonal constraints

The general quadratically constrained quadratic program (QQP) is an important modelling tool for many diverse problems. The QQP is in general NP hard, and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations and can be solved efficiently. For several special cases of QQP, the Lagrangian relaxation provides the exact optimal value. This means that there is a zero duality gap and the problem is tractable. It is important to know for which cases this is true, since they can then be used as subproblems to improve Lagrangian relaxation for intractable QQPs. In this paper we study the special QQP with orthogonal (matrix) constraints XX T= I. If C=0, the zero duality gap result holds if the redundant orthogonal constraints X T X= I are added. We show that this is not true in the general case. However, we show how to close the duality gap in the pure linear case by adding variables in addition to constraints.