Discrete Variable Problems Research Articles

Gate-Based Variational Quantum Algorithm for Truss Structure Size Optimization Problem

Quantum computing has become a pivotal innovation in computational science, offering novel avenues for tackling the increasingly complex and high-dimensional optimization challenges inherent in engineering design. This paradigm shift is particularly pertinent in the domain of structural optimization, where the intricate interplay of design variables and constraints necessitates advanced computational strategies. In this vein, the gate-based variational quantum algorithm utilizes quantum superposition and entanglement to improve search efficiency in large solution spaces. This paper delves into the gate-based variational quantum algorithm for the discrete variable truss structure size optimization problem. By reformulating this optimization challenge into a quadratic, unconstrained binary optimization framework, we bridge the gap between the discrete nature of engineering optimization tasks and the quantum computational paradigm. A detailed algorithm is outlined, encompassing the translation of the truss optimization problem into the quantum problem, the initialization and iterative evolution of a quantum circuit tailored to this problem, and the integration of classical optimization techniques for parameter tuning. The proposed approach demonstrates the feasibility and potential of quantum computing to transform engineering design and optimization, with numerical experiments validating the effectiveness of the method and paving the way for future explorations in quantum-assisted engineering optimizations.

Optimal Placement of Vibration Sensors for Industrial Robots Based on Bayesian Theory

This paper presents an optimal sensor placement method for vibration signal acquisition in the field of industrial robot health monitoring and fault diagnosis. Based on the general formula of Bayes and relative entropy, the evaluation function of sensor placement is deduced, and the modal confidence matrix is used to express the redundancy of sensor placement. The optimal placement of vibration sensors is described as a discrete variable optimization problem, which is defined as whether the existing sensor layout can obtain joint state information efficiently. The initial layout of the sensor was obtained from the structural simulation results of the industrial robots, and the initial layout was optimized by the derived objective function. The efficiency of the optimized layout in capturing joint state information is proven by the validation experiment with a simulation model. The problem of popularizing the optimization method in engineering is solved by a verification experiment without a simulation model. The optimal sensor placement method provides a theoretical basis for industrial robots to acquire vibration data effectively.

Research on Assembly Scheduling of Complex Electromechanical Products Based on Improved Particle Swarm Optimization

The assembly shop production scheduling of complex electromechanical products is a typical discrete variable NP-hard problem. In order to solve the assembly job scheduling problem of a large transformer assembly shop, an improved particle swarm algorithm is used to solve the optimal assembly task scheduling result. In order to minimize the maximum completion time, the traditional particle swarm optimization (PSO) algorithm was improved according to the characteristics of large transformer assembly process. First, the two-stage coding method of process and assembly team is adopted. Then, in order to improve the quality of the initial solution, the initialization method that minimizes the completion time is selected, and the adaptive inertia weight optimization speed update formula is adopted. Finally, the simulation study is carried out with relevant example data. Compared with the traditional PSO, the maximum completion time and convergence speed are significantly improved, which verifies the effectiveness of the improved PSO, and obtains the production scheduling Gantt chart of the workshop.

Solving binary semidefinite programming problems and binary linear programming problems via multi-objective programming

‎In recent years the binary quadratic program has grown in‎ ‎combinatorial optimization‎. ‎Quadratic programming can‎ ‎be formulated as a semidefinite programming problem‎. ‎In this paper‎, ‎we consider the general form of‎ ‎binary semidefinite programming problems (BSDP)‎.‎ ‎We show the optimal solutions of the BSDP belong to the efficient set of a semidefinite multiobjective programming problem (SDMOP)‎. ‎Although‎ ‎finding all efficient points for multiobjective is not an easy problem‎, ‎but‎ ‎solving a continuous problem would be easier than a discrete variable problem‎. ‎In this paper‎, ‎we solve an SDMOP‎, ‎as an auxiliary‎, ‎instead of BSDP‎. We show the performance of our method by generating and solving random problems.

An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures

Consider the scattering of a time-harmonic elastic plane wave by a bi-periodic rigid surface. The displacement of elastic wave motion is modeled by the three-dimensional Navier equation in an unbounded domain above the surface. Based on the Dirichlet-to-Neumann (DtN) operator, which is given as an infinite series, an exact transparent boundary condition is introduced and the scattering problem is formulated equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed to solve the discrete variational problem where the DtN operator is truncated into a finite number of terms. The a posteriori error estimate takes account of the finite element approximation error and the truncation error of the DtN operator which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.

Design of curved composite panels for maximum buckling load using sequential permutation search algorithm

New buckling optimal solutions for curved composite panels are presented in terms of principle of virtual work and sequential permutation search (SPS) algorithm. By utilizing the principle of virtual displacements, the buckling load is calculated and the accuracy of the derived buckling solutions is verified by comparison with experimental data and numerical results. SPS utilizes the inherent mechanical properties of composite laminates in which the bending stiffness of the outer layers is more sensitive than those close to the mid-surface. Curved composite panels exhibit a variety of length-to-arc ratios and radius-to-length ratios, under axial compression, pure shear, and combined loadings, are optimized. Some of the results are compared with the global optima obtained by enumeration method, and SPS shows good robustness in locating the global optima. In addition, the buckling load carrying capacity of optimized curved composite panels can be improved by as much as twice over cross-ply laminates. Low computational costs of SPS algorithm imply its potential for solving high-dimensional discrete variable optimization problems of composite structures.

Linear Integrable Systems on Quad-Graphs

Abstract In the first part of the paper, we classify linear integrable (multidimensionally consistent) quad-equations on bipartite isoradial quad-graphs in $\mathbb{C}$, enjoying natural symmetries and the property that the restriction of their solutions to the black vertices satisfies a Laplace type equation. The classification reduces to solving a functional equation. Under certain restriction, we give a complete solution of the functional equation, which is expressed in terms of elliptic functions. We find two real analytic reductions, corresponding to the cases when the underlying complex torus is of a rectangular type or of a rhombic type. The solution corresponding to the rectangular type was previously found by Boutillier, de Tilière, and Raschel. Using the multidimensional consistency, we construct the discrete exponential function, which serves as a basis of solutions of the quad-equation. In the second part of the paper, we focus on the integrability of discrete linear variational problems. We consider discrete pluri-harmonic functions, corresponding to a discrete two-form with a quadratic dependence on the fields at black vertices only. In an important particular case, we show that the problem reduces to a two-field generalization of the classical star-triangle map. We prove the integrability of this novel 3D system by showing its multidimensional consistency. The Laplacians from the first part come as a special solution of the two-field star-triangle map.

Cohort intelligence with self-adaptive penalty function approach hybridized with colliding bodies optimization algorithm for discrete and mixed variable constrained problems

Recently, several socio-/bio-inspired algorithms have been proposed for solving a variety of problems. Generally, they perform well when applied for solving unconstrained problems; however, their performance degenerates when applied for solving constrained problems. Several types of penalty function approaches have been proposed so far for handling linear and non-linear constraints. Even though the approach is quite easy to understand, the precise choice of penalty parameter is very much important. It may further necessitate significant number of preliminary trials. To overcome this limitation, a new self-adaptive penalty function (SAPF) approach is proposed and incorporated into socio-inspired Cohort Intelligence (CI) algorithm. This approach is referred to as CI–SAPF. Furthermore, CI–SAPF approach is hybridized with Colliding Bodies Optimization (CBO) algorithm referred to as CI–SAPF–CBO algorithm. The performance of the CI–SAPF and CI–SAPF–CBO algorithms is validated by solving discrete and mixed variable problems from truss structure domain, design engineering domain, and several problems of linear and nonlinear in nature. Furthermore, the applicability of the proposed techniques is validated by solving two real-world applications from manufacturing engineering domain. The results obtained from CI–SAPF and CI–SAPF–CBO are promising and computationally efficient when compared with other nature inspired optimization algorithms. A non-parametric Wilcoxon’s rank sum test is performed on the obtained statistical solutions to examine the significance of CI–SAPF–CBO. In addition, the effect of the penalty parameter on pseudo-objective function, penalty function and constrained violations is analyzed and discussed along with the advantages over other algorithms.

Mixed-integer second-order cone optimization for composite discrete ply-angle and thickness topology optimization problems

Discrete variable topology optimization problems are usually solved by using solid isotropic material with penalization (SIMP), genetic algorithms (GA), or mixed-integer nonlinear optimization (MINLO). In this paper, we propose formulating discrete ply-angle and thickness topology optimization problems as a mixed-integer second-order cone optimization (MISOCO) problem. Unlike SIMP and GA methods, MISOCO efficiently finds the problem’s globally optimal solution. Furthermore, in contrast with existing MISOCO formulations of discrete ply-angle optimization problems, our reformulations allow the structure to change topology, consider the more realistic Tsai–Wu stress yield criteria constraint, and eliminate checkerboard patterns using simple linear constraints. We address two types of discrete ply-angle and thickness problems: a structural mass minimization problem and a compliance optimization problem where the objective is to maximize the structural stiffness. For each element, one first chooses if the element is present or not in the structure. One can then choose the element’s ply-angle and thickness from a finite set of possibilities for the former case. The discrete design space for ply-angle and thickness is a result of manufacturing limitations. To improve the problem’s MISOCO solution approach, we develop valid inequality constraints to tighten the continuous relaxation of the MISOCO reformulation. We compare the performance of various MISOCO solvers: Gurobi, CPLEX, and MOSEK to solve the MISOCO reformulation. We also use BARON to solve the original MINLO formulations of the problems. Our results show that solving the MISOCO problem’s formulation using MOSEK is the most efficient solution approach.

Further elaborations on topology optimization via sequential integer programming and Canonical relaxation algorithm and 128-line MATLAB code

This paper provides further elaborations on discrete variable topology optimization via sequential integer programming and Canonical relaxation algorithm. Firstly, discrete variable topology optimization problem for minimum compliance subject to a material volume constraint is formulated and approximated by a sequence of discrete variable sub-programming with the discrete variable sensitivity. The differences between continuous variable sensitivity and discrete variable sensitivity are discussed. Secondly, the Canonical relaxation algorithm designed to solve this sub-programming is presented with a discussion on the move limit strategy. Based on the discussion above, a compact 128-line MATLAB code to implement the new method is included in Appendix 1. As shown by numerical experiments, the 128-line code can maintain black-white solutions during the optimization process. The code can be treated as the foundation for other problems with multiple constraints.

The problem of Lagrange in discrete field theory

A discrete Lagrange problem in a fibered manifold on a cellular complex is defined as a Lagrangian density and a constraint submanifold in the 1-jets space of the manifold. After defining the concepts of admissible section and infinitesimal admissible variation, the aim of these problems is to find admissible sections that are critical for the action functional of the Lagrangian density with respect to the infinitesimal admissible variations. For admissible sections satisfying a certain regularity condition, we prove that such critical sections are the solutions of an extended unconstrained discrete variational problem canonically associated to the problem of Lagrange (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established for the discrete Lagrange problems considered. Finally, the whole theory is illustrated with an example of geometric interest: the discrete harmonic 1-forms on the elliptic or hyperbolic discrete plane, for which we obtain two explicit variational integrators: one with initial conditions (Cauchy problem) and other with boundary conditions (Dirichlet problem).

Entropy-controlled Last-Passage Percolation

We introduce a natural generalization of Hammersley’s Last-Passage Percolation (LPP) called Entropy-controlled Last-Passage Percolation (E-LPP), where points can be collected by paths with a global (path-entropy) constraint which takes into account the whole structure of the path, instead of a local ($1$-Lipschitz) constraint as in Hammersley’s LPP. Our main result is to prove quantitative tail estimates on the maximal number of points that can be collected by a path with entropy bounded by a prescribed constant. The E-LPP turns out to be a key ingredient in the context of the directed polymer model when the environment is heavy-tailed, which we consider in (Berger and Torri (2018)). We give applications in this context, which are essentials tools in (Berger and Torri (2018)): we show that the limiting variational problem conjectured in (Ann. Probab. 44 (2016) 4006–4048), Conjecture 1.7, is finite, and we prove that the discrete variational problem converges to the continuous one, generalizing techniques used in (Comm. Pure Appl. Math. 64 (2011) 183–204; Probab. Theory Related Fields 137 (2007) 227–275).

Topology optimization via sequential integer programming and Canonical relaxation algorithm

The mathematical essence of structural topology optimization is large-scale nonlinear integer programming. To overcome its huge computational burden, a popular way is to relax the 0–1 variable constraints and transform the integer programming problem into a continuous variable programming problem which can be solved by using gradient based mathematical programming methods. To cope with the variable transformation, the well-known SIMP (Solid Isotropic Material with Penalty) method introduces interpolation schemes for the material properties versus design variables with penalty and achieves great success and popularity. However, there is no doubt that directly tackling the large-scale nonlinear integer programming is very important. This paper solves the structural topology optimization problems with single or multiple constraints by applying the Canonical Dual Theory (CDT) by Gao and Ruan (2010) together with Sequential Approximate Programming approach under the classic structural topology optimization formulations.Following the Sequential Approximate Programming (SAP) frame from the structural optimization, the present paper firstly utilizes sensitivity information to construct the explicit and separable approximate Sequential Quadratic Integer Programming (SQIP) or Sequential Linear Integer Programming (SLIP) subproblems for the topology optimization. And then, the subproblems are solved by applying the Canonical relaxation algorithm based on CDT theory. Their special mathematical structures are exploited to develop analytic solution of Kuhn–Tucker condition of the dual programming. Numerical experiments of two linear and quadratic integer programming problems with random coefficients assert that the Canonical relaxation algorithm can obtain approximate solutions with good properties very efficiently and the dual gap is negligible when the number of design variables increases.Because move limit strategies play a key role in many search algorithms of structural optimization, this paper combines one of two different move limit strategies within the new method. The new method first solves a set of classic topology optimization problems with only material usage constraint, including minimum structural compliance design under constant load, maximum heat transfer efficiency for the heat conduction problem. And then we apply the method to the topology optimization problems with multiple constraints, including minimum structural compliance design under an additional local displacement constraint and minimum structural compliance design under infill constraints. The results of these problems demonstrate that the new method can efficiently solve the discrete variable structural topology optimization problems with multiple nonlinear constraints or many local linear constraints in a unified and systematic way. Beyond that, the new method can achieve integer solutions when combined with the move limit strategy of controlling volume fraction parameter. It can deal with much more design variables than the general branch-and-bound method and makes no use of any sensitivity threshold or heuristic stabilization scheme during the iterative process in comparison with BESO method. Finally, the new method in this paper can be further developed as a general solver for these large-scale discrete variable structural topology optimization problems.

Topology optimization of pretensioned reflector antennas with unified cable-bar model

From innovative high-rise buildings through spacecraft, applications of prestressed cables are not uncommon. But the use of prestressed cables in ground-based reflector antennas is very rare. In this paper, prestressed cables were introduced to the ground-based antennas to achieve the lightweight design requirements. To determine the layout of the prestressed cables, a gradient-based topology optimization model was established. In this optimization model, the continuously varying 'lack of fit' is used to replace the discrete element type variables as the design variables, and smooth sigmoid functions are introduced to unify the discrete allowable stress of cable and bar elements. Eventually, the discrete topological variable optimization problem is transformed into a continuous one. Besides, to avoid the occurrence of grey elements in the optimized results, a parameter control strategy for the sigmoid functions is carried out during the optimization procedure. As an example, the optimizations of an 8 m truss antenna and an 8 m cable-truss antenna are implemented, and the discussions of the results are given.

School based optimization algorithm for design of steel frames

In this paper, a school-based optimization (SBO) algorithm is applied to the design of steel frames. The objective is to minimize total weight of steel frames subjected to both strength and displacement requirements specified by the American Institute of Steel Construction (AISC) Load Resistance Factor Design (LRFD). SBO is a metaheuristic optimization algorithm inspired by the traditional educational process that operates within a multi-classroom school. SBO is a collaborative optimization strategy, which allows for extensive exploration of the search space and results in high-quality solutions. To investigate the efficiency of SBO algorithm, several popular benchmark frame examples are optimized and the designs are compared to other optimization methods in the literature. Results indicate that SBO can develop superior low-weight frame designs when compared to other optimization methods and improves computational efficiency in solving discrete variable structural optimization problems.

The application of improved adaptive genetic algorithm in the optimization of discrete variables

In view of the problem of optimization design of complex discrete variables in practical application, the optimization design method of discrete variables and the processing methods of discrete variables are analyzed. The shortcomings of these methods are summarized, and the characteristics of genetic algorithms and discrete variables are combined to propose a self-adaptive discrete non-uniform crossover operator and adaptive discrete mapping mutation operator, so that the genetic operation can be implemented in discrete space for higher efficiency search. Combined with the optimal individual protection strategy and adaptive crossover and mutation strategy execution sequence, accelerate the population rate of global optimization, overall improve the efficiency of the algorithm of discrete optimization, through the design of a practical discrete variable optimization problems research, verified the effectiveness of the algorithm to solve the problem of discrete optimization design.

Cohort intelligence algorithm for discrete and mixed variable engineering problems

In this study, the performance of an emerging socio inspired metaheuristic optimization technique referred to as Cohort Intelligence (CI) algorithm is evaluated on discrete and mixed variable nonlinear constrained optimization problems. The investigated problems are mainly adopted from discrete structural optimization and mixed variable mechanical engineering design domains. For handling the discrete solution variables, a round off integer sampling approach is proposed. Furthermore, in order to deal with the nonlinear constraints, a penalty function method is incorporated. The obtained results are promising and computationally more efficient when compared to the other existing optimization techniques including a Multi Random Start Local Search algorithm. The associated advantages and disadvantages of CI algorithm are also discussed evaluating the effect of its two parameters namely the number of candidates, and sampling space reduction factor.

Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels

In this manuscript we propose the discrete versions for the recently introduced fractional derivatives with nonsingular Mittag-Leffler function. The properties of such fractional differences are studied and the discrete integration by parts formulas are proved. Then a discrete variational problem is considered with an illustrative example. Finally, some more tools for these derivatives and their discrete versions have been obtained.

Symmetric duality for left and right Riemann–Liouville and Caputo fractional differences

A discrete version of the symmetric duality of Caputo–Torres, to relate left and right Riemann–Liouville and Caputo fractional differences, is considered. As a corollary, we provide an evidence to the fact that in case of right fractional differences, one has to mix between nabla and delta operators. As an application, we derive right fractional summation by parts formulas and left fractional difference Euler–Lagrange equations for discrete fractional variational problems whose Lagrangians depend on right fractional differences.

Probability collectives for solving discrete and mixed variable problems

Probability collectives for solving discrete and mixed variable problems