An Agent Based Evolutionary Approach for Nonlinear Optimization with Equality Constraints

Targeting the adjustment of the errors-in-variables (EIV) model with equality and inequality constraints, a general solution that is similar to the classical least square adjustment is proposed based on the penalty function and the weight in measurement. Firstly, we take the equality constraints as inequality constraints that do not satisfy the constraint conditions and construct the penalty functions of equality and inequality constraints, respectively. Thus, the inequality constrained optimization problem is transformed into an unconstrained optimization problem. Then the detailed calculation formula and approximate accuracy evaluation formula of the general solution are deduced. The iteration formula of the general solution is easy regarding comprehension and applicable in implementation. It can not only solve the EIV model with equality and inequality constraints respectively, but also address the EIV model with equality and inequality constraints simultaneously. In addition, it can promote the Gauss–Markov (G-M) model with equality and inequality constraints. Finally, three examples (i.e., equality constraints, inequality constraints and those with equality and inequality constraints) are validated, indicating that the general solution is effective and feasible. The results show that the general solution is effective and feasible.

A variable reduction strategy for evolutionary algorithms handling equality constraints

The performance of evolutionary algorithms can be heavily undermined when\nconstraints limit the feasible areas of the search space. For instance, while\nCovariance Matrix Adaptation Evolution Strategy is one of the most efficient\nalgorithms for unconstrained optimization problems, it cannot be readily\napplied to constrained ones. Here, we used concepts from Memetic Computing,\ni.e. the harmonious combination of multiple units of algorithmic information,\nand Viability Evolution, an alternative abstraction of artificial evolution, to\ndevise a novel approach for solving optimization problems with inequality\nconstraints. Viability Evolution emphasizes elimination of solutions not\nsatisfying viability criteria, defined as boundaries on objectives and\nconstraints. These boundaries are adapted during the search to drive a\npopulation of local search units, based on Covariance Matrix Adaptation\nEvolution Strategy, towards feasible regions. These units can be recombined by\nmeans of Differential Evolution operators. Of crucial importance for the\nperformance of our method, an adaptive scheduler toggles between exploitation\nand exploration by selecting to advance one of the local search units and/or\nrecombine them. The proposed algorithm can outperform several state-of-the-art\nmethods on a diverse set of benchmark and engineering problems, both for\nquality of solutions and computational resources needed.\n

Adaptive modeling strategy for constrained global optimization with application to aerodynamic wing design

In this paper, the classical exact absolute value penalty function method is used for solving a new class of nonconvex nonsmooth optimization problems. Nonconvex nondifferentiable optimization problems with both inequality and equality constraints are considered here, in which not all functions constituting them have the fundamental property of convex functions and most classes of generalized convex functions—namely, a stationary point of such a function is its global minimum. It is proved for such nonconvex optimization problems that there exists a finite threshold of penalty parameters equal to the largest absolute value of a Lagrange multiplier such that, for every penalty parameter exceeding this lower bound, there is the equivalence between an optimal solution in the original constrained minimization problem with ‐invex functions and a minimizer in its associated penalized optimization problem with the exact l1 penalty function. Further, under nondifferentiable ‐invexity assumptions, a characterization of a saddle point of the Lagrange function, defined for the considered constrained optimization problem in terms of minimizers of its associated exact penalized optimization problem with the exact l1 penalty function, is presented.

This paper presents a new method for solving a linear programming problem, which is an extended version of the one previously presented by the author. The optimal solution of a linear programming problem is composed of some inequality constraints in their equality form. Then, it is possible to recognize the problem of finding the equality constraints which constitute the optimal solution. Some index is introduced to estimate the equality constraints among the inequality conditions. The estimated constraints give some solution which may be optimal or not optimal but feasible solution, or infeasible solution. Then the remains of the algorithm are the methods to transfer the infeasible/feasible solution to a feasible/optimal solution. Some candidate algorithm with a new optimality criterion is proposed in this paper and is confirmed its validity by many examples. The algorithm is also applied for solving a traveling salesperson problem.

A multi-objective evolutionary algorithm for steady-state constrained multi-objective optimization problems

Penalty and partitioning techniques to improve performance of QUBO solvers

A new and flexible method for the geometric modeling of ship hull forms is presented. The underlying methodology is the parametric design of B-spline curves and surfaces. Important form parameters like displacement, center of buoyancy, waterplane area, center of flotation etc. are utilized as high-level descriptors of the intended shapes. Instead of interactively manipulating B-spline vertices, the generation process is viewed as a constrained optimization problem where fairness measures are applied as objective functions, vertices are treated as design variables and form parameters are preserved as equality constraints - making the approach a novelty in B­spline modeling. The new design methodology is discussed and mathematical principles are outlined. Examples are given to demonstrate the applicability of the parametric approach. They include the design of a 33ft IMS yacht with focus on the bare hull without rudder and keel.

On the equality constraints tolerance of Constrained Optimization Problems

This work studies transportation, assignment, transshipment, shortest path, and dynamic lot sizing problems. Particularly, it shows how to modify their formulations to handle excess supply and node elimination. The paper proceeds by using the conservation of flow to derive the original constraints of these problems. Thereafter, a general form of the conservation of flow is used to reformulate these problems. The results are problems with inequality constraints. This paper shows that solving each of these new formulations returns the same optimal solution as its original form. It also shows how problems with inequality constraints can handle excess supply and node elimination. Along the way, some additional properties of these alternative formulations are explored.

As outlined in chapter 1, integrated optimum structural design represents an optimization problem with both equality constraints and inequality constraints. In this method the analysis and optimal design of the structure are carried out simultaneously. The disadvantage of this approach, however, is that the formulation results in a large size problem. Generalized geometric programming is a powerful mathematical tool that can be used to solve the multidimensional optimization problems with nonlinear equations, Burns (1985). In this chapter we will present the generalized geometric programming (GGP) and its extensions to the solution of problems with equality constraints (GGPE). We will also describe the IGGP solver, Burns (1982), and the added extensions for the optimal solution.

Finding cause-effect relationships is the central aim of many studies in the physical, behavioral, social and biological sciences. We consider two well-known mathematical causal models: Structural equation models and causal Bayesian networks. When we hypothesize a causal model, that model often imposes constraints on the statistics of the data collected. These constraints enable us to test or falsify the hypothesized causal model. The goal of our research is to develop efficient and reliable methods to test a causal model or distinguish between causal models using various types of constraints. For linear structural equation models, we investigate the problem of generating a small number of constraints in the form of zero partial correlations, providing an efficient way to test hypothesized models. We study linear structural equation models with correlated errors focusing on the graphical aspects of the models. We provide a set of local Markov properties and prove that they are equivalent to the global Markov property. For causal Bayesian networks, we study equality and inequality constraints imposed on data and investigate a way to use these constraints for model testing and selection. For equality constraints, we formulate an implicitization problem and show how we may reduce the complexity of the problem. We also study the algebraic structure of the equality constraints. For inequality constraints, we present a class of inequality constraints on both nonexperimental and interventional distributions.

The present paper is devoted to the application of the space transformation technique for solving linear and nonlinear programming problems. By using a surjective mapping the initial constrained optimization problem is transformed to a problem in a new space with only equality constraints. For the numerical solution of the latter problem we use the generalized gradient-projection method and Newton’s method. After inverse transformation to the initial space we obtain the family of numerical methods for solving optimization problems with equality and inequality constraints. In the linear programming case after some simplification and after choosing a particular exponential transformation function we obtain from the proposed methods the well-known primal-dual interior point linear programming algorithm and the so-called “variation of Kermarkar’s algorithm”.

A new multi-section based technique for constrained optimization problems with interval-valued objective function