Standard Optimization Problem Research Articles

MPCC: strong stability of weakly nondegenerate S-stationary points

ABSTRACT In this paper, we consider the class of mathematical programmes with complementarity constraints (MPCC). Specifically, we focus on strong stability of M- and S-stationary points for MPCC. Kojima introduced this concept for standard nonlinear optimization problems. It refers to several well-posedness properties of the underlying problem. Besides its topological definition, the challenge is to state an algebraic characterization of strong stability. We obtain such a description for S-stationary points whose components of Lagrange vectors corresponding to bi-active constraints do not mutually vanish. We call these points weakly nondegenerate. Moreover, we show that a particular constraint qualification is necessary for strong stability.

A population-based DNN-augmented optimization method for designing truss structures

Truss design problems have been extensively investigated for various purposes. How to efficiently optimize truss structures and obtain diverse optimized designs is still a challenging task. In this paper, a population-based DNN (deep neural network) -augmented method is proposed to address this issue. In the proposed method, truss design problems are firstly transformed into a DNN-based design loss minimization problem. The DNN model is gradually generated in an adaptive updating manner, and is used to indicate the truss design problem. In addition, the transformed minimization problem can be efficiently solved by DNN computation and a population of trusses can be optimized accordingly. Through the investigation of four truss design problems (two standard optimization problems and two optimization problems considering the design diversity), the effectiveness of the proposed method is validated. In addition, three aspects affecting the performance of the proposed method are also investigated, and the robustness and applicability of the proposed method are demonstrated.

Multi-objective trajectory planning for industrial robots using a hybrid optimization approach

Abstract In this paper, a hybrid approach organized in four phases is proposed to solve the multi-objective trajectory planning problem for industrial robots. In the first phase, a transcription of the original problem into a standard multi-objective parametric optimization problem is achieved by adopting an adequate parametrization scheme for the continuous robot configuration variables. Then, in the second phase, a global search is performed using a population-based search metaheuristic in order to build a first approximation of the Pareto front (PF). In the third phase, a local search is applied in the neighborhood of each solution of the PF approximation using a deterministic algorithm in order to generate new solutions. Finally, in the fourth phase, results of the global and local searches are gathered and postprocessed using a multi-objective direct search method to enhance the quality of compromise solutions and to converge toward the true optimal PF. By combining different optimization techniques, we intend not only to improve the overall search mechanism of the optimization strategy but also the resulting hybrid algorithm should keep the robustness of the population-based algorithm while enjoying the theoretical properties of convergence of the deterministic component. Also, the proposed approach is modular and flexible, and it can be implemented in different ways according to the applied techniques in the different phases. In this paper, we illustrate the efficiency of the hybrid framework by considering different techniques available in various numerical optimization libraries which are combined judiciously and tested on various case studies.

Using the Cuckoo Optimization to Initialize the Harmony Memory in Harmony Search Algorithm to Find New Hybrid (CSHS) Algorithm

The current paper seeks to present a hybrid harmony search algorithm, using CS and HS. The proposed algorithm takes advantage of using a method to initialize the harmony memory (HM) with the help of the cuckoo search optimization algorithm. The HS algorithm is inspired by the method used by musicians to create and enhance harmony in music. It follows 3 main principles of pitch adjustment (PA), HM consideration (HMC), and random selection (RS). The performance of HS can be affected by its poor accuracy in optimization and speed of convergence, 2 main issues with it. However, to improve the HS algorithm, Cuckoo search can be used as it can do a local search using one parameter only, other than the size of the population, making it work more efficiently. This current method has been tested for its validity and efficiency in performance by being implemented on many international standard optimization problems. The results confirm the fact that the performance of the currently proposed algorithm is better and more efficient in finding solutions compared to HS and other algorithms.

Метод эффективных вычитаний: работа с данными Иркутского радара некогерентного рассеяния

For incoherent scatter measurements, the effective subtraction technique is to alternate the duration of amplitude-modulated signals between a pair of consequently radiated pulses. The resulting gain of spatial resolution enables us to steadily assess the electron density profile by the Faraday rotation method. The paper describes the electron density measurement technique, which involves analyzing narrow-band signals from Irkutsk Incoherent Scatter Radar, and proposes an automated method of determining the electron density for the problem in which the convolution of the radiated signal waveform with backscatter signal cannot be neglected. The inverse problem of electron density recovery is considered as a standard nonlinear optimization problem, which is solved using the algorithms for global and local optimization applied consequently. We compare the electron density profiles obtained by analyzing different pulse waveforms and from Irkutsk ionosonde data.

Effective subtraction technique: implementation for Irkutsk Incoherent Scatter Radar

For incoherent scatter measurements, the effective subtraction technique is to alternate the duration of amplitude-modulated signals between a pair of consequently radiated pulses. The resulting gain of spatial resolution enables us to steadily assess the electron density profile by the Faraday rotation method. The paper describes the electron density measurement technique, which involves analyzing narrow-band signals from Irkutsk Incoherent Scatter Radar, and proposes an automated method of determining the electron density for the problem in which the convolution of the radiated signal waveform with backscatter signal cannot be neglected. The inverse problem of electron density recovery is considered as a standard nonlinear optimization problem, which is solved using the algorithms for global and local optimization applied consequently. We compare the electron density profiles obtained by analyzing different pulse waveforms and from Irkutsk ionosonde data.

Generalized Nash equilibrium problems with mixed-integer variables

AbstractWe consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido–Isoda function. To any given instance of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.

A Direct Yaw Moment Control Framework Through Robust T-S Fuzzy Approach Considering Vehicle Stability Margin

The direct yaw moment control system of distributed drive electric vehicles provides a flexible control scheme to enhance the vehicle stability through four independently driven in-wheel motors. However, the vehicle nonlinearities may lead to the invalidation of control. To this end, a Takagi–Sugeno (T-S) fuzzy-based robust <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H∞</i> control method is proposed to ensure the vehicle performance while addressing the nonlinear challenge. First, thanks to the T-S fuzzy modeling technology, the tire nonlinear characteristics are described by fuzzy rules, based on which the vehicle lateral dynamics model is established. Next, the safety region represented by the tire slip angles phase plane is presented to evaluate the vehicle stability performance. Considering the control priority of vehicle handling performance and stability control with different stability margins, a multiobjective optimization function is transformed into a standard robust performance optimization problem with dynamic weight coefficients. A T-S fuzzy-based robust <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H∞</i> state feedback controller is then designed to ensure the system stability and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H∞</i> performance. Finally, the hardware-in-the-loop tests are conducted to validate the proposed controller. Comparative results show the effectiveness to improve the vehicle handling performance while ensuring the stability.

Operator norm-based determination of failure probability of nonlinear oscillators with fractional derivative elements subject to imprecise stationary Gaussian loads

An approximate analytical technique is developed for bounding the first-passage probability of lightly damped nonlinear and hysteretic oscillators endowed with fractional derivative elements and subjected to imprecise stationary Gaussian loads. In particular, the statistical linearization and stochastic averaging methodologies are integrated with an operator norm-based approach to formulate a numerically efficient proxy for the first-passage probability. This proxy is employed to determine the realizations of the interval-valued parameters of the excitation model that yield the extrema of the failure probability function. Ultimately, each failure probability bound is determined in a fully decoupled manner by solving a standard optimization problem followed by a single evaluation of the first-passage probability. The proposed approximate technique can be construed as an extension of a recently developed operator norm scheme to account for oscillators with fractional derivative elements. In addition, it can readily treat a wide range of nonlinear and hysteretic behaviors. To illustrate the applicability and effectiveness of the proposed technique, a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative elements subject to imprecise stationary Gaussian loads are considered as numerical examples.

Walrus optimizer: A novel nature-inspired metaheuristic algorithm

Metaheuristic algorithms are intelligent optimization approaches that lead the searching procedure through utilizing exploitation and exploration. The increasing complexity of real-world optimization problem has prompted the development of more metaheuristic algorithms. Hence, this work proposes a novel swarm intelligence algorithm, Walrus optimizer (WO). It is inspired by the behaviors of walruses that choose to migrate, breed, roost, feed, gather and escape by receiving key signals (danger signals and safety signals). To test the capability of the proposed algorithm, 23 standard functions and the benchmark suite from the IEEE (Institute of Electrical and Electronics Engineers) Congress on Evolutionary Computation (CEC) 2021 are used. In addition, to evaluate the practicability of the proposed algorithm to solve various real-world optimization problems, 6 standard classical engineering optimization problems are examined and compared. For statistical purposes, 100 independent optimization runs are conducted to determine the statistical measurements, including the mean, standard deviation, and the computation time of the program, by considering a predefined stopping criterion. Some well-known statistical analyses are also used for comparative purposes, including the Friedman and Wilcoxon analysis. The results demonstrate that the proposed algorithm can provide special stability features and very competitive performance in dealing with high-dimensional benchmarks and real-world problems. The proposal of WO promotes the continuous development and application expansion of artificial intelligence, improves the efficiency of optimization calculation, and provides powerful tools for solving complex problems in the real world. The source code of WO is publicly availabe at https://ww2.mathworks.cn/matlabcentral/fileexchange/154702-walrus-optimizer-wo.

A Low-Cost Alternating Projection Approach for a Continuous Formulation of Convex and Cardinality Constrained Optimization

We consider convex constrained optimization problems that also include a cardinality constraint. In general, optimization problems with cardinality constraints are difficult mathematical programs which are usually solved by global techniques from discrete optimization. We assume that the region defined by the convex constraints can be written as the intersection of a finite collection of convex sets, such that it is easy and inexpensive to project onto each one of them (e.g., boxes, hyper-planes, or half-spaces). Taking advantage of a recently developed continuous reformulation that relaxes the cardinality constraint, we propose a specialized penalty gradient projection scheme combined with alternating projection ideas to compute a solution candidate for these problems, i.e., a local (possibly non-global) solution. To illustrate the proposed algorithm, we focus on the standard mean-variance portfolio optimization problem for which we can only invest in a preestablished limited number of assets. For these portfolio problems with cardinality constraints, we present a numerical study on a variety of data sets involving real-world capital market indices from major stock markets. In many cases, we observe that the proposed scheme converges to the global solution. On those data sets, we illustrate the practical performance of the proposed scheme to produce the effective frontiers for different values of the limited number of allowed assets.

On the structure of optimal solutions in a mathematical programming problem in a convex space

We consider an optimization problem in a convex space E with an affine objective function, subject to J affine constraints, where J is a given nonnegative integer. We apply the Feinberg-Shwartz lemma in finite dimensional convex analysis to show that there exists an optimal solution, which is in the form of a convex combination of no more than J+1 extreme points of E. The concerned problem does not seem to fit into the framework of standard convex optimization problems.

Model Predictive Control for Residential Battery Storage System: Profitability Analysis

For increased penetration of energy production from renewable energy sources at a utility scale, battery storage systems (BSSs) are a must. Their levelized cost of electricity (LCOE) has drastically decreased over the last decade. Residential battery storage, mostly combined with photovoltaic (PV) panels, also follow this falling prices trend. The combined effect of the COVID-19 pandemic and the war in Ukraine has caused such a dramatic increase in electricity prices that many consumers have adjusted their strategies to become prosumers and self-sufficient as feed-in subsidies continue to drop. In this study, an investigation is conducted to determine how profitable it is to install BSSs in homes with regards to battery health and the levelized cost of total managed energy. This is performed using mixed-integer linear programming (MILP) in MATLAB, along with its embedded solver Intlinprog. The results show that a reasonable optimized yearly cycling rate of the BSS can be reached by simply considering a non-zero cost for energy cycling through the batteries. This cost is simply added to the electricity cost equation of standard optimization problems and ensures a very good usage rate of the batteries. The proposed control does not overreact to small electricity price variations until it is financially worth it. The trio composed of feed-in tariffs (FITs), electricity costs, and the LCOE of BSSs represents the most significant factors. Ancillary grid service provision can represent a substantial source of revenue for BSSs, besides FITs and avoided costs.

Sparse conic reformulation of structured QCQPs based on copositive optimization with applications in stochastic optimization

AbstractRecently, Bomze et al. introduced a sparse conic relaxation of the scenario problem of a two stage stochastic version of the standard quadratic optimization problem. When compared numerically to Burer’s classical reformulation, the authors showed that there seems to be almost no difference in terms of solution quality, whereas the solution time can differ by orders of magnitudes. While the authors did find a very limited special case, for which Burer’s reformulation and their relaxation are equivalent, no satisfying explanation for the high quality of their bound was given. This article aims at shedding more light on this phenomenon and give a more thorough theoretical account of its inner workings. We argue that the quality of the outer approximation cannot be explained by traditional results on sparse conic relaxations based on positive semidenifnite or completely positive matrix completion, which require certain sparsity patterns characterized by chordal and block clique graphs respectively, and put certain restrictions on the type of conic constraint they seek to sparsify. In an effort to develop an alternative approach, we will provide a new type of convex reformulation of a large class of stochastic quadratically constrained quadratic optimization problems that is similar to Burer’s reformulation, but lifts the variables into a comparatively lower dimensional space. The reformulation rests on a generalization of the set-completely positive matrix cone. This cone can then be approximated via inner and outer approximations in order to obtain upper and lower bounds, which potentially close the optimality gap, and hence can give a certificate of exactness for these sparse reformulations outside of traditional, known sufficient conditions. Finally, we provide some numerical experiments, where we asses the quality of the inner and outer approximations, thereby showing that the approximations may indeed close the optimality gap in interesting cases.

Ordinal optimization through multi-objective reformulation

We analyze combinatorial optimization problems with ordinal, i.e., non-additive, objective functions that assign categories (like good, medium and bad) rather than cost coefficients to the elements of feasible solutions. We review different optimality concepts for ordinal optimization problems and discuss their similarities and differences. We then focus on two prevalent optimality concepts that are shown to be equivalent. Our main focus lies on the investigation of a bijective linear transformation that transforms ordinal optimization problems to associated standard multi-objective optimization problems with binary cost coefficients. Since this transformation preserves all properties of the underlying problem, problem-specific solution methods remain applicable. A prominent example is dynamic programming and Bellman’s principle of optimality, that can be applied, e.g., to ordinal shortest path and ordinal knapsack problems. We investigate the interrelation between scalarization techniques and methods based on the hypervolume indicator when applied to the ordinal and the transformed problem, respectively. Furthermore, we extend our results to multi-objective optimization problems that combine ordinal and real-valued objective functions.

Sunflower optimization algorithm for multi-strategy-assist parameter identification of solar cell models

A novel optimization method, namely, the elite opposition learning and polynomial steps-based sunflower optimization (EOPSFO) algorithm, has been proposed to solve engineering problems. To speed up the convergence, the elite opposition-based learning and polynomial steps strategy is applied to automatically determine the search step adapted in each iteration. To verify the performance of EOPSFO, the feasibility of EOPSFO is first verified using various benchmarking and some standard optimization problems. In addition, EOPSFO is used to determine the parameters of the single diode model and double diode model. Results show that EOPSFO can be regarded as a competitive algorithm in optimization problems.

A Convex Formulation for Robust Control of Bilinear Positive Systems With Evolutionary Dynamics

In this article, optimal control of bilinear positive systems with uncertain evolutionary dynamics is studied. The first objective is to determine optimal control signals such that the effect of the disturbance signals on the outputs of the nominal dynamics is minimized in terms of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{H}_2$</tex-math> </inline-formula> norm performance index. Properties of the positive systems, Metzler matrices, and the exploitable structure of the problem are utilized to show that the underlying problem is representable as a standard convex optimization problem. Next, the optimal strategy is examined in the presence of polytopic uncertainties. In this case, the optimization problem is a semi-infinite min-max problem. It is shown that under some assumptions, the underlying problem is equivalent to a finite min-max problem, i.e., it suffices to check only the vertices of the polytopic set for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{H}_2$</tex-math> </inline-formula> norm minimization. The proposed approach for polytopic bilinear positive systems is non-iterative, does not involve parameter tuning or nonsmooth optimization algorithms, and can be solved using off-the-shelf solvers. The effectiveness of the proposed approach is investigated for the optimal combination therapy of human immunodeficiency virus (HIV).

Sum Rate Optimization Scheme of UAV-Assisted NOMA under Hardware Impairments

In the unmanned aerial vehicle (UAV) assisted non-orthogonal multiple access (NOMA) networks, the practical hardware impairments (HIs) and resource allocation is still a challenging problem. Most existing research on resource allocation algorithms for UAV communication is considered with the ideal hardware condition. However, the impact of HIs on system performance cannot be ignored, especially in the case of high bit rates. Considering the HIs, most studies are from the perspective of performance analysis. The resource allocation of UAV relay-assisted NOMA systems is investigated in this paper with HIs. We aim to maximize the sum rate by jointly optimizing the deployment of UAV and transmit power. To address this problem, we first transformed the mixed integer programming problem (MIPP) into a standard convex optimization problem based on successive convex approximation (SCA) technology. Then, we introduced the Lagrangian dual transformation and quadratic transform methods to solve the power allocation problem. Finally, we propose an effective iterative algorithm to achieve an approximate optimal solution. Numerical results demonstrate that the proposed algorithm achieved better performance in terms of the sum rate compared with other benchmark schemes.

Improved Anomaly Detection by Using the Attention-Based Isolation Forest

A new modification of the isolation forest called the attention-based isolation forest (ABIForest) is proposed for solving the anomaly detection problem. It incorporates an attention mechanism in the form of Nadaraya–Watson regression into the isolation forest to improve the solution of the anomaly detection problem. The main idea underlying the modification is the assignment of attention weights to each path of trees with learnable parameters depending on the instances and trees themselves. Huber’s contamination model is proposed to be used to define the attention weights and their parameters. As a result, the attention weights are linearly dependent on learnable attention parameters that are trained by solving a standard linear or quadratic optimization problem. ABIForest can be viewed as the first modification of the isolation forest to incorporate an attention mechanism in a simple way without applying gradient-based algorithms. Numerical experiments with synthetic and real datasets illustrate that the results of ABIForest outperform those of other methods. The code of the proposed algorithms has been made available.

On the extended Choquet-Sugeno-like operator

In this short paper, we introduce and discuss the extended Choquet-Sugeno-like operator. It generalizes most of modifications of the Choquet integral on a finite set known in the literature providing a partially positive answer to the question posed by H. Bustince et al. in 2021. It also includes unknown Choquet-like and Sugeno-like operators. The standard optimization problem in cluster analysis is formulated using multivariable extension of the new operator.