This study presents a method for form-finding and analysis of hyperelastic tensegrity structures based on a special strut finite element and unconstrained nonlinear programming. The strut element can function as a hyp...
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This study presents a method for form-finding and analysis of hyperelastic tensegrity structures based on a special strut finite element and unconstrained nonlinear programming. The strut element can function as a hyperelastic truss element with an initial cut in its undeformed length or as a strut element that shows constant force irrespectively of its nodal displacements. For the hyperelastic strut element, the invariants of the Right Cauchy-Green deformation tensor are written in terms of the element's nodal displacements and the cut in the element's undeformed length. The structure's total potential energy is expressed as function of its nodal displacements and the cuts in the elements' undeformed lengths. The minimization of this function is a nonlinear programming problem where the displacements are the unknowns. The form-finding procedure is performed by a static analysis where the stiffness matrix maybe singular along the path to equilibrium without causing convergence problems. The mathematical model includes the element's cross-sectional deformation while the element moves in space, fully modelling its three-dimensional character. The constraint for incompressibility is satisfied exactly, eliminating the need for a penalty or augmented Lagrangian method.
The ant colony optimization algorithm (ACOA) is hybridized with nonlinear programming (NLP) for the optimal design of sewer networks. The resulting problem is a highly constrained mixed integer nonlinear problem (MINL...
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The ant colony optimization algorithm (ACOA) is hybridized with nonlinear programming (NLP) for the optimal design of sewer networks. The resulting problem is a highly constrained mixed integer nonlinear problem (MINLP) presenting a challenge even to the modern heuristic search methods. In the proposed hybrid method, The ACOA is used to determine pipe diameters while the NLP is used to determine the pipe slopes of the network by proposing two different formulations. In the first formulation, named ACOA-NLP1 a penalty method is used to satisfy the problem constraints while in the second one, named ACOA-NLP2, the velocity and flow depth constraints are expressed in terms of the slope constraints which are easily satisfied as box constraint of the NLP solver leading to a considerable reduction of the search space size. In addition, the assumption of minimum cover depth at the network inlets is used to calculate the nodal cover depths and the pump and drop heights at the network nodes, if required, leading to a complete solution. The total cost of the constructed solution is used as the objective function of the ACOA, guiding the ant toward minimum cost solutions. Proposed hybrid methods are used to solve three test examples, and the results are presented and compared with those produced by a conventional application of ACOA. The results indicate the effectiveness and efficiency of the proposed formulations and in particular the ACOA-NLP2 to optimally solve the sewer network design optimization problems. (C) 2018 Water Environment Federation
The finite-dimensional McCormick second-order sufficiency theory for nonlinear programming problems with a finite number of constraints is now a classical part of the optimization literature. It was introduced by McCo...
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The finite-dimensional McCormick second-order sufficiency theory for nonlinear programming problems with a finite number of constraints is now a classical part of the optimization literature. It was introduced by McCormick in 1967 and an improved version was given by Fiacco and McCormick in their 1968 award-winning book. Later it was learned that in 1953 Pennisi had presented exactly the same theory. Many authors, most notably Maurer and Zowe in a widely cited paper in 1978, argue that the Pennisi-McCormick theory cannot be extended to infinite dimensions without adding further assumptions, by producing a counterexample. They then extend the theory to infinite dimensions, allowing for an infinite number of constraints, by strengthening the sufficient conditions required. In the current paper we use a fundamental principle for second-order sufficiency to extend the Pennisi-McCormick second-order theory as stated in R-n to infinite-dimensional normed vector spaces, without strengthening the conditions. The Maurer and Zowe infinite-dimensional counterexample carried an infinite number of constraints. Hence they seemed to be unaware that the extension of the Pennisi-McCormick theory to infinite dimensions was possible provided the original feature of a finite number of constraints was maintained.
Water allocation under limited water supplies is becoming more important as water becomes scarcer. Optimization models are frequently used to provide decision support to enhance water allocation under limited water su...
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Water allocation under limited water supplies is becoming more important as water becomes scarcer. Optimization models are frequently used to provide decision support to enhance water allocation under limited water supplies. Correct modelling of the underlying soil-moisture balance calculations at the field scale, which governs optimal allocation of water is a necessity for decision-making. Research shows that the mathematical programming formulation of soil-moisture balance calculations presented by Ghahraman and Sepaskhah (2004) may malfunction under limited water supplies. A new model formulation is presented in this research that explicitly models deep percolation and evapotranspiration as a function of soil-moisture content. The new formulation also allows for the explicit modelling of inefficiencies resulting from nonuniform irrigation. Modelling inefficiencies are key to the evaluation of the economic profitability of deficit irrigation. Ignoring increasing efficiencies resulting from deficit irrigation may render deficit irrigation unprofitable. The results show that ignoring increasing efficiencies may overestimate the impact of deficit irrigation on maize yields by a maximum of 2.2 tons per hectare.
Magnetometer is a significant sensor for integrated navigation. However, it suffers from many kinds of unknown dynamic magnetic disturbances. We study the problem of online estimating such disturbances via a nonlinear...
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Magnetometer is a significant sensor for integrated navigation. However, it suffers from many kinds of unknown dynamic magnetic disturbances. We study the problem of online estimating such disturbances via a nonlinear optimization aided by the intermediate quaternion estimation from inertial fusion. The proposed optimization is constrained by the geographical distribution of the magnetic field forming a constrained nonlinear programming. The uniqueness of the solution has been verified mathematically, and we design an interior-point-based solver for efficient computation on embedded chips. It is claimed that the designed scheme mainly outperforms in dealing with the challenging bias estimation problem under static motion as the previous representatives can hardly achieve. Experimental results demonstrate the effectiveness of the proposed scheme on high accuracy, fast response, and low computational load.
We argue that reducing nonlinear programming problems to a simple canonical form is an effective way to analyze them, specially when the gradients of the constraints are linearly dependent. To illustrate this fact, we...
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We argue that reducing nonlinear programming problems to a simple canonical form is an effective way to analyze them, specially when the gradients of the constraints are linearly dependent. To illustrate this fact, we solve an open problem about constraint qualifications using this canonical form.
Failure to satisfy Constraint Qualifications (CQs) leads to serious convergence difficulties for state-of-the-art nonlinear programming (NLP) solvers. Since this failure is often overlooked by practitioners, a strateg...
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Failure to satisfy Constraint Qualifications (CQs) leads to serious convergence difficulties for state-of-the-art nonlinear programming (NLP) solvers. Since this failure is often overlooked by practitioners, a strategy to enhance the robustness properties for problems without CQs is vital. Inspired by penalty merit functions and barrier-like strategies, we propose and implement a combination of both in Ipopt. This strategy has the advantage of consistently satisfying the Linear Independence Constraint Qualification (LICQ) for an augmented problem, readily enabling regular step computations within the interior-point framework. Additionally, an update rule inspired by the work of Byrd et al. (2012) is implemented, which provides a dynamic increase of the penalty parameter as stationary points are approached. Extensive test results show favorable performance and robustness increases for our ℓ 1 —penalty strategies, when compared to the regular version of Ipopt. Moreover, a dynamic optimization problem with nonsmooth dynamics formulated as a Mathematical Program with Complementarity Constraints (MPCC) was solved in a single optimization stage without additional reformulation. Thus, this ℓ 1 — strategy has proved useful for a broad class of degenerate NLPs.
Homotopy methods are powerful tools for solving nonlinear programming. Their global convergence can be generally established under conditions of the nonemptiness and boundness of the interior of the feasible set, the ...
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Homotopy methods are powerful tools for solving nonlinear programming. Their global convergence can be generally established under conditions of the nonemptiness and boundness of the interior of the feasible set, the Positive Linear Independent Constraint Qualification (PLICQ), which is equivalent to the Mangasarian-Fromovitz Constraint Qualification (MFCQ), and the normal cone condition. This paper provides a comparison of the existing normal cone conditions used in homotopy methods for solving inequality constrained nonlinear programming.
This paper presents several didactic examples of the nonlinear programming (NLP) problems solved with Mathematica. We solved examples of Karush-Kuhn-Tucker necessary conditions, Lagrange multipliers method, convex opt...
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This paper presents several didactic examples of the nonlinear programming (NLP) problems solved with Mathematica. We solved examples of Karush-Kuhn-Tucker necessary conditions, Lagrange multipliers method, convex optimization, and graphical method. We compared the hand calculation in Karush-Kuhn-Tucker method with Lagrange multipliers method. The paper contains Mathematica symbolic codes used for Karush-Kuhn-Tucker necessary conditions and the Hessian analysis in convex optimization. We present also some didactic graphs for various aspects of NLP problems using plots and dynamic plots. The use of Mathematica during teaching students about NLP by Computer Algebra System (CAS) seems to be very useful both as the calculations support (checking hand calculation) and when creating didactic graphical visualizations using dynamic plots. We did not find in available literature any similar example of NLP problems solved with CAS or the use of dynamic plots.
Inspired by classical sensitivity results for nonlinear optimization, we derive and discuss new quantitative bounds to characterize the solution map and dual variables of a parametrized nonlinear program. In particula...
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