AI News, More from r/MachineLearning
- On Sunday, June 3, 2018
- By Read More
More from r/MachineLearning
Coming up with a completely brand new way of doing something that isn't somehow very related or built upon current methods is actually quite difficult.
So, to answer your question - to develop a really very novel algorithm you need to get good at math and play with data and solve problems and find a problem you can solve, and apply the intuition you've gained and the math you've learned to develop a solution.
To assemble a bunch of parts as in the 3rd example, or hack together an idea as in the 2nd (though Dropout does come with theoretical results, so I wouldn't call their paper a hack - but I could easily see the same solution being developed in a hackish manner) just takes lots of practice and understanding the tools / models you are using.
- On Monday, June 4, 2018
- By Read More
Mathematical Problems in Engineering
An effective hybrid algorithm is proposed for solving multiobjective optimization engineering problems with inequality constraints.
To improve the computational efficiency and maintain rapid convergence, a cautious BFGS iterative format is utilized to approximate the Hessian matrices of the objective functions instead of evaluating them exactly.
This proposed strategy has been applied to the crashworthiness design of the primary energy absorption device’s crash box structure and front rail under low-speed frontal collision.
Multiobjective optimization problems (MOPs) are encountered in many fields, such as energy systems [1, 2], management , structural optimum design [4, 5], and scheduling [6, 7].
For instance, for obtaining robust optimal solutions, a multiobjective optimization algorithm that combines particle swarm optimization and the gradient search technique was introduced , and Sun et al.
To obtain a faster convergence rate and improve the stability of convergence and efficiency for solving engineering design problems, numerical algorithms gradually attracted increased attention from researchers [26, 27].
The scalarization technique, which was first introduced by Geoffrion , has been widely used to transform a multiobjective problem into a scalar one, and new insights into the weighted sum method have been reported in .
Peng and Wang  established an effective adaptive surrogate for solving transfer trajectory optimization, the computing speed of which was almost 8 times faster than that of directly solving the actual model.
To solve MOPs in practical engineering projects, an effective optimization strategy is proposed based on surrogated modeling and applied to crashworthiness optimization of the main absorption devices of the automobile.
In general, a multiobjective optimization problem with inequality constraints can be expressed aswhere is a vector function, is the number of single-objective functions, and is the decision variable vector, which consists of several variables that are generated in the feasible region and satisfy the inequality constraint for any To describe the optimal solutions of the MOP accurately and clearly, some definitions are clearly stated in the following section.
The Pareto front (denoted as ) consists of all corresponding values of function against all Pareto-optimal solutions and To properly adapt the numerous effective unconstrained algorithms, the original problem (1) is converted into an unconstrained problem based on the Frisch interior penalty function.
First, a penalty item is established as follows:where is a penalty coefficient, which is updated adaptively by (5) and denotes the inequality constraint with .where is a constant parameter.
Based on the above processes, problem (1) has been converted into an unconstrained problem, which can be described asTo effectively solve the unconstrained problem, the preferred method is the BFGS quasi-Newton algorithm, which is superior to other similar methods, such as Newton’s method and the DFP method.
The Taylor expansion of an unconstrained scalar function near can be expressed aswhere and are the gradient function and Hessian matrix, respectively, of target function at .
Next, inequality (8) can be converted toConsider is the function with respect to the variable and the function is denoted asAssume that is the solution the minimum problem (11).If is a nonsingular matrix, the solution is obtained as (12) by solving problem (11).The BFGS quasi-Newton’s method is an optimization method that uses (12) as its search direction.
Generally, the positive-definite symmetric matrix is preferred to be similar in value to the Hessian matrix , which satisfies the equation in approximation (13).in which the vectors and satisfy the conditionEquation (14) is known as both the quasi-Newton equation and the quasi-Newton condition.
In this section, the weighted sum method is utilized to transform the vector function into a scalar function, which is presented aswhere are weighting factors that meet the condition for and can be divided into various sets according to the practical design demands.
Based on the abovementioned processes, the original multiobjective problem with inequality constraints has been converted to an unconstrained scalar problem, which is stated as In this section, a cautious BFGS quasi-Newton algorithm (CBQNA) for the MOP based weighted sum method is proposed.
First, based on the interior penalty function method and the weighted sum technique, a quasi-Newton direction for the MOP is obtained by solvingIn this paper, the approximation matrix is updated by a cautious BGFS equation (18), which was introduced in  and can guarantee the positive definiteness of after each iteration.where and are positive constant parameters;
If this condition is not satisfied, the gradient direction will be replaced by another search direction, which is expressed asThen, the classical Armijo rule is used to determine a proper step length along quasi-Newton’s search direction .
Furthermore, the positive definiteness of can be preserved by utilizing updating equation (18) and setting , and the full step length along the descent search direction is usually obtained according to the study .
The proposed algorithm is primarily utilized to solve MOPs in engineering projects, the mathematical models of which are usually replaced by approximate convex functions that can ensure that the global optimal solution of the problem be achieved.
Particularly, the proposed algorithm is more effective in solving engineering problems than algorithms such as Newton’s method, because it avoids computing second-order information of the objective functions, which does not exist in the general engineering case.
Particularly, when determining the descent search direction, the proposed algorithm avoids computing the Hessian matrices of objective functions, which is of great significance for solving practical engineering problems.
To describe how to construct sample points and determine the optimal number of samples by orthogonal design and CCD, the procedural details of the two design methods can be separately summarized as follows.
The eight points that are located at the summit of the cube are of the same experimental format, the six sample points that are located at the end of the center axis are of the same type, and the center point in red is a sample that needs to be repeated six times.
Therefore, considering the characteristics of the proposed algorithm, the establishment of a linear, quadratic, or cubic polynomial surface is favored for describing the relationships between objective responses and design variables.
Similarly, a quadratic polynomial model is usually constructed as in expression (26) and the cubic polynomial surface can be obtained in the same manner.For simplicity, expression (26) can be converted to a linear form as follows:where and denote items with and , respectively.
To accurately estimate the precision of the approximated models, the correlation coefficient and adjusted index need to be calculated according to formulas (28) and (29), respectively.where denotes the number of factors and , are the predictive value and the actual value, respectively, of the average measurement of .
To effectively simulate the performance of the crash box under a low-speed impact, one end of each crash box has been fully constrained, while the other end is subject to front collision by a rigid wall of infinite mass at a low speed of 4 m/s along the x-axis, as shown in Figure 5.
Thus, improving energy absorption and reducing maximal impact force have been selected as simultaneous objectives, while the thicknesses of the four plates are chosen as design variables.
Consequently, an MOP of the crash box that is based on crashworthiness is mathematically expressed aswhere and are two objective functions of the vector , which consists of four decision variables, which are subject to boundary condition .
In addition, the approximate mode of the constraint function is presented in expression (33).After calculation, the coefficients of determination of energy absorption, maximal impact force, and mass constraint are 98.60%, 99.11%, and 1, respectively, while the adjusted coefficients of determination are 95.6%, 97.11%, and 1, respectively.
Thus, problem (30) can be converted into a standard mathematical problem, which is expressed as follows:where and are the maximal value of the energy absorption and average value of the maximal collision force, respectively.
Hence, the mathematical model for multiobjective optimization of the front rail, which consists of two objective functions in terms of three design variables, is constructed aswhere is the vector of design variables;
Following the novel optimization strategy’s procedures, orthogonal design is utilized to generate 25 design sampling points (i.e., four levels for each of the three design variables), which are clearly presented in Table 3 with their corresponding responses.
According to the 25 plans of the DOE, which are shown in Table 3, the quadratic response surface models of the energy absorption and maximal impact force responses were constructed in expressions (36) and (37), respectively.
The values of coefficient for the energy absorption and collision force reach 99% and 98%, respectively, and the correction coefficient values are 99% and 95%, respectively.
Thus, problem (35) is converted into a standard biobjective optimization problem with inequality constraints, which is expressed as follows:where and 234.9 kN are the average value of energy absorption and the maximal collision force for the 25 groups, respectively.
After the simulation by software LS-DYNA, the energy absorption and collision force responses, the specific energy absorption (SEA), and the mass of the structure, compared with the original design, are presented in Table 5.
The calculation time of the optimization of the S-shaped front rail is approximately 32.8 s for 200 groups’ optimal solutions, which demonstrates that the optimization strategy that is based on the proposed algorithm is effective and promising in solving complicated engineering problems in which the objective function’s second derivative is difficult to calculate or does not exist.
In our future work, evolutionary algorithms will be combined with the proposed algorithm to generate better initial value for definitely avoiding sinking into pseudo-optimal solution and the convergence of the proposed hybrid algorithm will be theoretically proven.
- On Monday, September 23, 2019
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