One-dimensional Optimization Research Articles

An efficient global optimization algorithm for maximizing the sum of two generalized Rayleigh quotients

Maximizing the sum of two generalized Rayleigh quotients (SRQ) can be reformulated as a one-dimensional optimization problem, where the function value evaluations are reduced to solving semi-definite programming (SDP) subproblems. In this paper, we first use the dual SDP subproblem to construct an explicit overestimation and then propose a branch-and-bound algorithm to globally solve (SRQ). Numerical results demonstrate that it is even more efficient than the recent SDP-based heuristic algorithm.

Optimal Unimodular Sequences Design Method for Active Sensing Systems

In active sensing systems, unimodular sequences with low autocorrelation sidelobes are widely adopted as modulation sequences to improve the distance resolution and antijamming performance. In this paper, in order to meet the requirements of specific practical engineering applications such as suppressing certain correlation coefficients and finite phase, we propose a new algorithm to design both continuous phase and finite phase unimodular sequences with a low periodic weighted integrated sidelobe level (WISL). With the help of the transformation matrix, such an algorithm decomposes the N-dimensional optimization problem into N one-dimensional optimization problems and then uses the iterative method to search the optimal solutions of the N one-dimensional optimization problems directly. Numerical experiments demonstrate the effectiveness and the convergence property of the proposed algorithm.

Constrained transmit beampattern design for colocated MIMO radar

This paper considers the constrained waveform design for Multiple-Input Multiple-Output (MIMO) radar to synthesize a desired beampattern. Specifically, resorting to SemiDefinite Programming (SDP) related technique, we first minimize Integration Sidelobe Level (ISL) to optimize the waveform covariance matrix enforcing a uniform elemental power restriction as well as a 3dB bandwidth constraint. Then, based on Least Square (LS) approach, we present the existing Cyclic Algorithm (CA) and a new Sequential Iterative Algorithm (SIA) to devise the waveform under a constant modulus constraint, and a similarity constraint to allow the designed waveform sharing the similarity feature with a given reference waveform. In particular, the proposed SIA directly optimizes the objective function and its each iteration turns the multidimensional optimization problem into multiple one-dimensional optimization problems with closed-form solutions. Finally, we assess the effectiveness of the proposed technique through numerical simulations in comparison with CA.

Quadratic Optimization With Similarity Constraint for Unimodular Sequence Synthesis

This paper considers unimodular sequence synthesis under similarity constraint for both the continuous and discrete phase cases. A computationally efficient iterative algorithm for the continuous phase case (IA-CPC) is proposed to sequentially optimize the quadratic objective function. The quadratic optimization problem is turned into multiple one-dimensional optimization problems with closed-form solutions. For the discrete phase case, we present an iterative block optimization algorithm. Specifically, we partition the design variables into $K$ blocks, and then, we sequentially optimize each block via exhaustive search while fixing the remaining $K-1$ blocks. Finally, we evaluate the computational costs and performance gains of the proposed algorithms in comparison with power method-like and semidefinite relaxation related techniques.

Convergence control of single loop approach for reliability-based design optimization

For solution of reliability-based design optimization (RBDO) problems, single loop approach (SLA) shows high efficiency. Thus SLA is extensively used in RBDO. However, the iteration solution procedure by SLA is often oscillatory and non-convergent for RBDO with nonlinear performance function. This prevents the application of SLA to engineering design problems. In this paper, the chaotic single loop approach (CLSA) is proposed to achieve the convergence control of original iterative algorithm in SLA. The modification involves automated selection of the chaos control factor by solving a novel one-dimensional optimization model. Additionally, a new oscillation-checking method is constructed to detect the oscillation of iterative process of design variables. The computational capability of CLSA is demonstrated through five benchmark examples and one stiffened shell application. The comparison of numerical results indicates that CSLA is more efficient than the double loop approach and the decoupled approach. CSLA also solves the RBDO problems with highly nonlinear performance function and non-normal random variables stably.

Parameter Estimation in Sensor Networks Under Probabilistic Censoring

We consider the problem of estimating an unknown parameter that is observed indirectly by a sensor network. Each of the sensors exhibits probabilistic nonlinear behavior causing censoring (clipping) of the signals at random. The signals then get transmitted over fading wireless channels to the fusion center. We develop three algorithms for estimating the unknown parameter: first is based on Pearson's Method of Moments, which involves the calculation of the statistical moments of the model that can be derive exactly; second is based on the Maximum-Likelihood Estimator , which results in an intractable likelihood function. To evaluate the likelihood function, we develop a novel approximation via a nonparametric probability density estimator that is based on series expansions of the Gram–Charlier family of basis functions; and third is the Marginalized Least Squares in which we solve the least squares problem by marginalizing out the random unknown nuisance parameters of the model. All three algorithms enjoy a low computational complexity, and only involve solving a one-dimensional optimization problem that is simple to implement in practice. We compare the performance of the proposed algorithms under various system configurations and show various tradeoffs between the algorithms as a function of system parameters (e.g., number of sensors, frame length, signal-to-noise ratio, sensing quality etc.).

Convergence conditions and numerical comparison of global optimization methods based on dimensionality reduction schemes

This paper is devoted to numerical global optimization algorithms applying several ideas to reduce the problem dimension. Two approaches to the dimensionality reduction are considered. The first one is based on the nested optimization scheme that reduces the multidimensional problem to a family of one-dimensional subproblems connected in a recursive way. The second approach as a reduction scheme uses Peano-type space-filling curves mapping multidimensional domains onto one-dimensional intervals. In the frameworks of both the approaches, several univariate algorithms belonging to the characteristical class of optimization techniques are used for carrying out the one-dimensional optimization. Theoretical part of the paper contains a substantiation of global convergence for the considered methods. The efficiency of the compared global search methods is evaluated experimentally on the well-known GKLS test class generator used broadly for testing global optimization algorithms. Results for representative problem sets of different dimensions demonstrate a convincing advantage of the adaptive nested optimization scheme with respect to other tested methods.

Real-Time Estimation of Sensorless Planar Robot Contact Information

[abstFig src='/00290003/11.jpg' width='300' text='Estimation of robot contact information' ] The real-time estimation of sensorless planar robot contact information is a very important but also difficult subject in human-robot interaction. This paper proposes a method for the real-time estimation of contact location and contact force along a planar joint robot manipulator without using external sensory systems. A momentum-based method is used to estimate external joint torques due to the contact force and to determine a minimum contact range firstly. A nonlinear constrained optimization algorithm is presented to search the contact point. The contact force is calculated by dynamics. The searching space determined by the momentum-based approach is limited within the length range of the contact arm, so the solution speed of the optimization algorithm is high. The proposed method of combining observation algorithm and optimization algorithm transforms a complex detection problem of the any contact point on the robot body into a simple one-dimensional optimization solution with simple bound. The effectiveness of the proposed approach is validated through simulations and experimental results for the planar robot manipulator.

Dual form reduction in the atomic optimization method

One-dimensional optimization problems with a polynomial objective function and polynomial matrix inequality constraints are considered. For problems dual to their linear relaxations, a transformation is presented that makes them compatible with the atomic optimization method, both in its basic and in the generalized form with a reduced number of atoms.

A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems

A new class of disturbance covariance matrix estimators for radar signal processing applications is introduced following a geometric paradigm. Each estimator is associated with a given unitary invariant norm and performs the sample covariance matrix projection into a specific set of structured covariance matrices. Regardless of the considered norm, an efficient solution technique to handle the resulting constrained optimization problem is developed. Specifically, it is shown that the new family of distribution-free estimators shares a shrinkagetype form; besides, the eigenvalues estimate just requires the solution of a one-dimensional convex problem whose objective function depends on the considered unitary norm. For the two most common norm instances, i.e., Frobenius and spectral, very efficient algorithms are developed to solve the aforementioned one-dimensional optimization leading to almost closed form covariance estimates. At the analysis stage, the performance of the new estimators is assessed in terms of achievable Signal to Interference plus Noise Ratio (SINR) both for a spatial and a Doppler processing assuming different data statistical characterizations. The results show that interesting SINR improvements with respect to some counterparts available in the open literature can be achieved especially in training starved regimes.

Constrained radar waveform design algorithm for spectral coexistence

The design problem of radar waveform to enhance target delectability while ensuring spectral compatibility is considered. Precisely, an iterative sequential optimisation algorithm is developed to jointly optimise signal-to-interference-plus-noise ratio and stopband frequency energy. In each iteration of the proposed algorithm, the multidimensional optimisation problem is turned into multiple one-dimensional optimisation problems with closed-form solutions. Finally, the computational cost and objective value of the proposed algorithm in comparison with the existing method is assessed.

Combined Method (1D + 3D) of the Axial Turbine’s Stage Aerodynamic Optimization

The paper presents the optimization method of turbine stages based on step-by-step application of onedimensional and three-dimensional optimization techniques of the main stage geometrical parameters. Using the developed method, the optimization of the 3rd stage of high pressure steam turbine K-540-23.5 was carried out. As a result of optimization a new stage with an absolute efficiency increase more than 1% compared to the original design was obtained. The reasons leading to such performance boost were analyzed and are represented in the paper.

Beam angle selection incorporation of anatomical heterogeneities for pencil beam scanning charged-particle therapy

In charged particle therapy with pencil beam scanning (PBS), localization of the dose in the Bragg peak makes dose distributions sensitive to lateral tissue heterogeneities. The sensitivity of a PBS plan to lateral tissue heterogeneities can be reduced by selecting appropriate beam angles. The purpose of this study is to develop a fast and accurate method of beam angle selection for PBS. The lateral tissue heterogeneity surrounding the path of the pencil beams at a given angle was quantified with the heterogeneity number representing the variation of the Bragg peak depth across the cross section of the beams using the stopping power ratio of body tissues with respect to water. To shorten the computation time, one-dimensional dose optimization was conducted along the central axis of the pencil beams as they were directed by the scanning magnets. The heterogeneity numbers were derived for all possible beam angles for treatment. The angles leading to the minimum mean heterogeneity number were selected as the optimal beam angle. Three clinical cases of head and neck cancer were used to evaluate the developed method. Dose distributions and their robustness to setup and range errors were evaluated for all tested angles, and their relation to the heterogeneity numbers was investigated. The mean heterogeneity number varied from 1.2 mm–10.6 mm in the evaluated cases. By selecting a field with a low mean heterogeneity number, target dose coverage and robustness against setup and range errors were improved. The developed method is simple, fast, accurate and applicable for beam angle selection in charged particle therapy with PBS.

Global Optimization on an Interval

This paper provides expressions for solutions of a one-dimensional global optimization problem using an adjoint variable which represents the available one-sided improvements up to the interval “horizon.” Interpreting the problem in terms of optimal stopping or optimal starting, the solution characterization yields two-point boundary problems as in dynamic optimization. Results also include a procedure for computing the adjoint variable, as well as necessary and sufficient global optimality conditions.

Analysis of Masked Data in a Series System Subjected to Sources of Shocks Under Type I Progressive Hybrid Censoring

Abstract In this paper, we consider the statistical analysis of masked data based on the fatal shock model under type I progressive hybrid censoring. It is assumed that the time of shock follows Weibull distribution. Maximum-likelihood estimators for the unknown parameters are obtained by solving a one-dimensional optimization problem. Approximate maximum-likelihood estimators have been proposed based on a Taylor series expansion, and they have explicit expressions. In addition, the Bayesian approach, combined with Gibbs sampling, are developed based on the assumption that the shape parameter has a log-concave function, and for the given shape parameter, the scale parameters have Gamma-Dirichlet priors. Finally, Monte Carlo simulations are performed to compare the performances of the proposed methods under different progressive censoring schemes and masking levels.

Concurrent optimal allocation of distributed manufacturing resources using extended Teaching-Learning-Based Optimization

The optimal allocation of distributed manufacturing resources is a challenging task for supply chain deployment in the current competitive and dynamic manufacturing environments, and is characterised by multiple objectives including time, cost, quality and risk that require simultaneous considerations. This paper presents an improved variant of the Teaching-Learning-Based Optimisation (TLBO) algorithm to concurrently evaluate, select and sequence the candidate distributed manufacturing resources allocated to subtasks comprising the supply chain, while dealing with the trade-offs among multiple objectives. Several algorithm-specific improvements are suggested to extend the standard form of TLBO algorithm, which is only well suited for the one-dimensional continuous numerical optimisation problem well, to solve the two-dimensional (i.e. both resource selection and resource sequencing) discrete combinatorial optimisation problem for concurrent allocation of distributed manufacturing resources through a focused trade-off within the constrained set of Pareto optimal solutions. The experimental simulation results showed that the proposed approach can obtain a better manufacturing resource allocation plan than the current standard meta-heuristic algorithms such as Genetic Algorithm, Particle Swarm Optimisation and Harmony Search. Moreover, a near optimal resource allocation plan can be obtained with linear algorithmic complexity as the problem scale increases greatly.

Operational zones for comparing metaheuristic and deterministic one-dimensional global optimization algorithms

Univariate continuous global optimization problems are considered in this paper. Several widely used multidimensional metaheuristic global optimization methods–genetic algorithm, differential evolution, particle swarm optimization, artificial bee colony algorithm, and firefly algorithm–are adapted to the univariate case and compared with three Lipschitz global optimization algorithms. For this purpose, it has been introduced a methodology allowing one to compare stochastic methods with deterministic ones by using operational characteristics originally proposed for working with deterministic algorithms only. As a result, a visual comparison of methods having different nature on classes of randomly generated test functions becomes possible. A detailed description of the new methodology for comparing, called “operational zones”, is given and results of wide numerical experiments with five metaheuristics and three Lipschitz algorithms are reported.

Efficient method for finding globally optimal solution of problem with weighted L p norm and L 2 norm objective function

This study proposes an iterative method to approximate an N-dimensional optimisation problem with a weighted L p and L 2 norm objective function by a sequence of N independent one-dimensional optimisation problems. Inspired by the existing weighted L 1 and L 2 norm separable surrogate functional (SSF) iterative shrinkage algorithm, there are N independent one-dimensional optimisation problems with weighted L p and L 2 norm objective functions. However, these optimisation problems are non-convex. Hence, they may have more than one locally optimal solutions and it is very difficult to find their globally optimal solutions. This paper proposes to partition the feasible set of each approximated problem into various regions such that the sign of the convexity of the objective function in each region remains unchanged. Here, there is no more than one stationary point in each region. By finding the stationary point in each region, the globally optimal solution of each approximated optimisation problem can be found. Besides, this study also shows that the sequence of the globally optimal solutions of the approximated problems converge to the globally optimal solution of the original optimisation problem. Computer numerical simulation results show that the proposed method outperforms the existing weighted L 1 and L 2 norm SSF iterative shrinkage algorithm.

Instantaneous magnitudes and instantaneous frequencies of signals with their positivity constraints via non‐smooth non‐convex functional constrained optimisation

This study proposes an iterative method to approximate an N-dimensional optimisation problem with a weighted Lp norm and L2 norm objective function by a sequence of N independent one-dimensional optimisation problems. This iterative method is inspired by the existing weighted L1 norm and L2 norm separable surrogate functional (SSF) iterative shrinkage algorithm. However, as these independent one-dimensional optimisation problems consist of weighted Lp norm and L2 norm objective functions, these optimisation problems are non-convex and they may have more than one locally optimal solutions. In general, it is very difficult to find their globally optimal solutions. To address this difficulty, this study proposes to partition the feasible set of each approximated problem into various regions such that the sign of the convexity of the objective function in each region remains unchanged. In this case, there is no more than one stationary point in each region. By finding the stationary point in each region, the globally optimal solution of each approximated optimisation problem can be found. Besides, this study also shows that the sequence of the globally optimal solutions of the approximated problems converge to the globally optimal solution of the original optimisation problem.

Sparse Penalized Forward Selection for Support Vector Classification

We propose a new binary classification and variable selection technique especially designed for high-dimensional predictors. Among many predictors, typically, only a small fraction of them have significant impact on prediction. In such a situation, more interpretable models with better prediction accuracy can be obtained by variable selection along with classification. By adding an ℓ1-type penalty to the loss function, common classification methods such as logistic regression or support vector machines (SVM) can perform variable selection. Existing penalized SVM methods all attempt to jointly solve all the parameters involved in the penalization problem altogether. When data dimension is very high, the joint optimization problem is very complex and involves a lot of memory allocation. In this article, we propose a new penalized forward search technique that can reduce high-dimensional optimization problems to one-dimensional optimization by iterating the selection steps. The new algorithm can be regarded as a forward selection version of the penalized SVM and its variants. The advantage of optimizing in one dimension is that the location of the optimum solution can be obtained with intelligent search by exploiting convexity and a piecewise linear or quadratic structure of the criterion function. In each step, the predictor that is most able to predict the outcome is chosen in the model. The search is then repeatedly used in an iterative fashion until convergence occurs. Comparison of our new classification rule with ℓ1-SVM and other common methods show very promising performance, in that the proposed method leads to much leaner models without compromising misclassification rates, particularly for high-dimensional predictors.