Unit Step Size Research Articles

Mixing enhancement with minimal thrust loss for the Mach 1.76 jet issuing from different beveled collar nozzles

The main objective of the present investigation is to enhance the jet mixing by introducing a nozzle collar exit inclination having bevel angles of 30°, 45°, and 60°. In addition to this, the present study also addresses the problem of investigating the jet which attains maximum mixing from beveled collar nozzle with the imposition of a minimal overall thrust loss penalty. The nozzles of the present study have been designed to deliver a uniform Mach 1.76 jet at the nozzle collar exit. The results thus obtained through numerical computations have been compared with that of the reference nozzle which has zero exit inclination. The investigations have been performed under both design and off-design conditions of the jet by varying the nozzle pressure ratio (NPR) from 4.5 to 6.5 with unit step size. It is seen that the rate of jet mixing is augmented with the increase in the bevel angle. Thus, the 60° beveled collar nozzle demonstrated the highest jet mixing in comparison to the reference nozzle, 30° and 45° beveled collar nozzle at each nozzle pressure ratio. However, 60° beveled collar nozzle has been substantially penalized with the highest thrust loss of about 6.5% at nozzle pressure ratio 4.5 and about 3.5% at nozzle pressure ratio 5.5 with respect to the reference nozzle and other nozzles of the present study. The 45° and 30° beveled collar nozzles reported the thrust penalty below 2.3% and 1%, respectively. The overall performance, which includes mixing enhancement along with the minimal thrust loss, has been found in the 45° beveled collar nozzle, which clearly justifies its preference over the other investigated nozzle configurations.

Distributed Optimization for Second-Order Discrete-Time Multiagent Systems With Set Constraints.

The optimization problem of second-order discrete-time multiagent systems with set constraints is studied in this article. In particular, the involved agents cooperatively search an optimal solution of a global objective function summed by multiple local ones within the intersection of multiple constrained sets. We also consider that each pair of local objective function and constrained set is exclusively accessible to the respective agent, and each agent just interacts with its local neighbors. By borrowing from the consensus idea, a projection-based distributed optimization algorithm resorting to an auxiliary dynamics is first proposed without interacting the gradient information of local objective functions. Next, by considering the local objective functions being strongly convex, selection criteria of step size and algorithm parameter are built such that the unique solution to the concerned optimization problem is obtained. Moreover, by fixing a unit step size, it is also shown that the optimization result can be relaxed to the case with just convex local objective functions given a properly chosen algorithm parameter. Finally, practical and numerical examples are taken to verify the proposed optimization results.

A Fast Temporal Decomposition Procedure for Long-Horizon Nonlinear Dynamic Programming

We propose a fast temporal decomposition procedure for solving long-horizon nonlinear dynamic programs. The core of the procedure is sequential quadratic programming (SQP) that utilizes a differentiable exact augmented Lagrangian as the merit function. Within each SQP iteration, we approximately solve the Newton system using an overlapping temporal decomposition strategy. We show that the approximate search direction is still a descent direction of the augmented Lagrangian provided the overlap size and penalty parameters are suitably chosen, which allows us to establish the global convergence. Moreover, we show that a unit step size is accepted locally for the approximate search direction and further establish a uniform, local linear convergence over stages. This local convergence rate matches the rate of the recent Schwarz scheme (Na et al. 2022). However, the Schwarz scheme has to solve nonlinear subproblems to optimality in each iteration, whereas we only perform a single Newton step instead. Numerical experiments validate our theories and demonstrate the superiority of our method. Funding: This work was supported by the National Science Foundation [Grant CNS-1545046] and the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research [Grant DE-AC02-06CH11347].

Improved RIP-based bounds for guaranteed performance of two compressed sensing algorithms

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 \({\delta _{3k}} < 1/\sqrt 3 \approx 0.5774\), 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 \({\delta _{3k}} < \left( {\sqrt 5 - 1} \right)/2 \approx 0.618\), and the bound for CoSaMP can be improved to δ4k < 0.5102.

Low‐complexity linear massive MIMO detection based on the improved BFGS method

Linear minimum mean square error (MMSE) detection achieves a good trade-off between performance and complexity for massive multiple-input multiple-output (MIMO) systems. To avoid the high-dimensional matrix inversion involved, MMSE detection can be transformed into an unconstrained optimization problem and then solved by efficient numerical algorithms in an iterative way. Three low-complexity Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton methods are proposed to iteratively realize massive MIMO MMSE detection without matrix inversion. The complexity can be reduced from O ( K 3 ) $\mathcal {O}(K^{3})$ to O ( L K 2 ) $\mathcal {O}(LK^{2})$ , where K and L denote the number of users and iterations, respectively. Leveraging the special properties of massive MIMO, the authors first explore a simplified BFGS method (named S-BFGS) to alleviate the computational burden in the search direction. For lower complexity, BFGS method with the unit step size (named U-BFGS) is presented subsequently. When the base station (BS)-to-user-antenna ratio (BUAR) is large enough, the two proposed BFGS methods can be integrated (named U-S-BFGS) to further reduce complexity. In addition, an efficient initialization strategy is devised to accelerate convergence. Simulation results verify that the proposed detection scheme can achieve near-MMSE performance with a small number of iterations L as low as 2 or 3.

Newton-type methods near critical solutions of piecewise smooth nonlinear equations

It is well-recognized that in the presence of singular (and in particular nonisolated) solutions of unconstrained or constrained smooth nonlinear equations, the existence of critical solutions has a crucial impact on the behavior of various Newton-type methods. On the one hand, it has been demonstrated that such solutions turn out to be attractors for sequences generated by these methods, for wide domains of starting points, and with a linear convergence rate estimate. On the other hand, the pattern of convergence to such solutions is quite special, and allows for a sharp characterization which serves, in particular, as a basis for some known acceleration techniques, and for the proof of an asymptotic acceptance of the unit stepsize. The latter is an essential property for the success of these techniques when combined with a linesearch strategy for globalization of convergence. This paper aims at extensions of these results to piecewise smooth equations, with applications to corresponding reformulations of nonlinear complementarity problems.

Unit stepsize for the Newton method close to critical solutions

As is well known, when initialized close to a nonsingular solution of a smooth nonlinear equation, the Newton method converges to this solution superlinearly. Moreover, the common Armijo linesearch procedure used to globalize the process for convergence from arbitrary starting points, accepts the unit stepsize asymptotically and ensures fast local convergence. In the case of a singular and possibly even nonisolated solution, the situation is much more complicated. Local linear convergence (with asymptotic ratio of 1/2) of the Newton method can still be guaranteed under reasonable assumptions, from a starlike, asymptotically dense set around the solution. Moreover, convergence can be accelerated by extrapolation and overrelaxation techniques. However, nothing was previously known on how the Newton method can be coupled in these circumstances with a linesearch technique for globalization that locally accepts unit stepsize and guarantees linear convergence. It turns out that this is a rather nontrivial issue, requiring a delicate combination of the analyses on acceptance of the unit stepsize and on the iterates staying within the relevant starlike domain of convergence. In addition to these analyses, numerical illustrations and comparisons are presented for the Newton method and the use of extrapolation to accelerate convergence speed.

A digital logic controlled variable resistor with isolation and scalability

This letter discusses an optocoupler based variable resistor circuit. The circuit has digital inputs and the output is a variable resistance with a resolution (unit step size) of 1 Ω. The circuit is easy to assemble especially for microsystems and low cost. Attractive features include variable resistance range scalability, galvanic isolation and digital logic based input.

Graininess of RGB-Display Space.

RGB–display space, that is, the ‘RGB–cube’, was sampled at 3,000 locations, uniformly and randomly distributed. Fifty observers contributed 60 samples each. At each location, participants synthesised a copy of the target, using a generic colour picker. The statistical distributions of errors as a function of location are used to define an overall measure of graininess. A smooth field of interpolated three-dimensional covariance ellipsoids represents an explicit, empirical Riemannian metric. The unit step size is about 20 times larger than the size of the classical MacAdam ellipses. We speculate that this metric might be found useful in various settings involving applications, because it reflects typical fuzziness encountered in generic tasks involving colour patterns such as images. Some of the more obvious applications are discussed.

An embedding of loop quantum cosmology in variables into a full theory context

Loop quantum cosmology in (b, v) variables, which is governed by a unit step size difference equation, is embedded into a full theory context based on similar variables. A full theory context here means a theory of quantum gravity arrived at using the quantisation techniques used in loop quantum gravity, however based on a different choice of elementary variables and classical gauge fixing suggested by loop quantum cosmology. From the full theory perspective, the symmetry reduction is characterised by the vanishing of certain phase space functions which are implemented as operator equations in the quantum theory. The loop quantum cosmology dynamics arise as the action of the full theory Hamiltonian on maximally coarse states in the kernel of the reduction constraints. An application of this reduction procedure to spherical symmetry is also sketched, with similar results, but only one canonical pair in (b, v) form.

Combining stabilized SQP with the augmented Lagrangian algorithm

For an optimization problem with general equality and inequality constraints, we propose an algorithm which uses subproblems of the stabilized SQP (sSQP) type for approximately solving subproblems of the augmented Lagrangian method. The motivation is to take advantage of the well-known robust behavior of the augmented Lagrangian algorithm, including on problems with degenerate constraints, and at the same time try to reduce the overall algorithm locally to sSQP (which gives fast local convergence rate under weak assumptions). Specifically, the algorithm first verifies whether the primal-dual sSQP step (with unit stepsize) makes good progress towards decreasing the violation of optimality conditions for the original problem, and if so, makes this step. Otherwise, the primal part of the sSQP direction is used for linesearch that decreases the augmented Lagrangian, keeping the multiplier estimate fixed for the time being. The overall algorithm has reasonable global convergence guarantees, and inherits strong local convergence rate properties of sSQP under the same weak assumptions. Numerical results on degenerate problems and comparisons with some alternatives are reported.

Conditional Gradient Algorithmsfor Rank-One Matrix Approximations with a Sparsity Constraint

The sparsity constrained rank-one matrix approximation problem is a difficult mathematical optimization problem which arises in a wide array of useful applications in engineering, machine learning, and statistics, and the design of algorithms for this problem has attracted intensive research activities. We introduce an algorithmic framework, called ConGradU, that unifies a variety of seemingly different algorithms that have been derived from disparate approaches, and that allows for deriving new schemes. Building on the old and well-known conditional gradient algorithm, ConGradU is a simplified version with unit step size that yields a generic algorithm which either is given by an analytic formula or requires a very low computational complexity. Mathematical properties are systematically developed and numerical experiments are given.

Inexact sequential quadratically constrained quadratic programming methods with nonmonotone line searches for convex programming

In this paper, we present inexact sequential quadratically constrained quadratic programming (SQCQP) methods with nonmonotone line searches for the convex programming problem. Kato, Narushima and Yabe proposed the SQCQP method whose subproblem is solved inexactly. Their inexact SQCQP method uses a monotone line search strategy. To reduce the number of merit function evaluations and to accept the unit step size easier, we apply the nonmonotone line search to their inexact SQCQP method. We present the algorithms of the inexact SQCQP method with the nonmonotone line searches and prove their global and superlinear convergence properties. Moreover, we give some numerical experiments to investigate the numerical performance of our proposed methods.

A perfect example for the BFGS method

Consider the BFGS quasi-Newton method applied to a general non-convex function that has continuous second derivatives. This paper aims to construct a four-dimensional example such that the BFGS method need not converge. The example is perfect in the following sense: (a) All the stepsizes are exactly equal to one; the unit stepsize can also be accepted by various line searches including the Wolfe line search and the Arjimo line search; (b) The objective function is strongly convex along each search direction although it is not in itself. The unit stepsize is the unique minimizer of each line search function. Hence the example also applies to the global line search and the line search that always picks the first local minimizer; (c) The objective function is polynomial and hence is infinitely continuously differentiable. If relaxing the convexity requirement of the line search function; namely, (b) we are able to construct a relatively simple polynomial example.

A Smoothing Trust Region Method for Ncps Based on the Smoothing Generalized Fischer-Burmeister Function

Based on a reformulation of the complementarity problem as a system of nonsmooth equations by using the generalized Fischer-Burmeister function, a smoothing trust region algorithm with line search is proposed for solving general (not necessarily monotone) nonlinear complementarity problems. Global convergence and, under a nonsingularity assumption, local Q-superlinear/Q-quadratic convergence of the algorithm are established. In particular, it is proved that a unit step size is always accepted after a finite number of iterations. Numerical results also confirm the good theoretical properties of our approach.

Proximal-like contraction methods for monotone variational inequalities in a unified framework II: general methods and numerical experiments

Approximate proximal point algorithms (abbreviated as APPAs) are classical approaches for convex optimization problems and monotone variational inequalities. In Part I of this paper (He et al. in Proximal-like contraction methods for monotone variational inequalities in a unified framework I: effective quadruplet and primary methods, 2010), we proposed a unified framework consisting of an effective quadruplet and a corresponding accepting rule. Under the framework, various existing APPAs can be grouped in the same class of methods (called primary or elementary methods) which adopt one of the geminate directions in the effective quadruplet and take the unit step size. In this paper, we extend the primary methods by using the same effective quadruplet and the accepting rule. The extended (general) contraction methods need only minor extra even negligible costs in each iteration, whereas having better properties than the primary methods in sense of the distance to the solution set. A set of matrix approximation examples as well as six other groups of numerical experiments are constructed to compare the performance between the primary (elementary) and extended (general) methods. As expected, the numerical results show the efficiency of the extended (general) methods are much better than that of the primary (elementary) ones.

Proximal-like contraction methods for monotone variational inequalities in a unified framework I: Effective quadruplet and primary methods

Approximate proximal point algorithms (abbreviated as APPAs) are classical approaches for convex optimization problems and monotone variational inequalities. To solve the subproblems of these algorithms, the projection method takes the iteration in form of u k+1=P ? [u k ?? k d k ]. Interestingly, many of them can be paired such that $\tilde{u}^{k} = P_{\varOmega}[u^{k} - \beta_{k}F(v^{k})] = P_{\varOmega}[\tilde {u}^{k} - (d_{2}^{k} - G d_{1}^{k})]$ , where inf?{β k }>0 and G is a symmetric positive definite matrix. In other words, this projection equation offers a pair of directions, i.e., $d_{1}^{k}$ and $d_{2}^{k}$ for each step. In this paper, for various APPAs we present a unified framework involving the above equations. Unified characterization is investigated for the contraction and convergence properties under the framework. This shows some essential views behind various outlooks. To study and pair various APPAs for different types of variational inequalities, we thus construct the above equations in different expressions according to the framework. Based on our constructed frameworks, it is interesting to see that, by choosing one of the directions ( $d_{1}^{k}$ and $d_{2}^{k}$ ) those studied proximal-like methods always utilize the unit step size namely ? k ?1.

Constrained Newton Methods for Transport Network Equilibrium Analysis

A set of constrained Newton methods were developed for static traffic assignment problems. The Newton formula uses the gradient of the objective function to determine an improved feasible direction scaled by the second-order derivatives of the objective function. The column generation produces the active paths necessary for each origin-destination pair. These methods then select an optimal step size or make an orthogonal projection to achieve fast, accurate convergence. These Newton methods based on the constrained Newton formula utilize path information to explicitly implement Wardrop's principle in the transport network modelling and complement the traffic assignment algorithms. Numerical examples are presented to compare the performance with all possible Newton methods. The computational results show that the optimal-step Newton methods have much better convergence than the fixed-step ones, while the Newton method with the unit step size is not always efficient for traffic assignment problems. Furthermore, the optimal-step Newton methods are relatively robust for all three of the tested benchmark networks of traffic assignment problems.

An % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % irpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaa % GcbaWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgaiuaacqWF % oe-taaa!44C5! $$ \mathcal{O} $$ (rL) infeasible interior-point algorithm for symmetric cone LCP via CHKS function

In this paper, we propose a theoretical framework of an infeasible interior-point algorithm for solving monotone linear complementarity problems over symmetric cones (SCLCP). The new algorithm gets Newton-like directions from the Chen-Harker-Kanzow-Smale (CHKS) smoothing equation of the SCLCP. It possesses the following features: The starting point is easily chosen; one approximate Newton step is computed and accepted at each iteration; the iterative point with unit stepsize automatically remains in the neighborhood of central path; the iterative sequence is bounded and possesses \( \mathcal{O} \)(rL) polynomial-time complexity under the monotonicity and solvability of the SCLCP.

Global line search strategies for nonlinear least squares problems based on curvature and projected curvature

In this article, we introduce global line search strategies based on the maximum projected curvature step (MPCS). This step is developed from the maximum curvature step (MCS), defined in [Chavent, G., 2004, Curvature steps and geodesic moves for nonlinear least squares descent algorithms. Inverse Problems in Science and Engineering.] and [Chavent, G., 2002, Curvature steps and geodesic moves for nonlinear least squares descent algorithms, INRIA Report.]. At a point in a descent curve, we introduce the optimization plan, on which we project the acceleration in order to obtain the projected curvature. For a search curve with bounded projected curvature, we compute the lower bound to the arc length of the first stationary point, which defines the MPCS. The convergence of the global strategies based on MPCSs and MCSs is studied for different common descent directions. Preliminary numerical results show that algorithms using the new strategies with steepest descent, Gauss–Newton or Levenberg–Marquardt direction are more efficient than the corresponding ones based on the unit step-size.