Nonlinear Programming

Kuhn, Harold W.; Tucker, A. W.

doi:10.1007/978-3-0348-0439-4_11

Cited by 305 publications

(248 citation statements)

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“…These two conditions were effectively combined by Hertz [64] and Signorini [122], together with the gap-contact alternative: either g > 0 and p = 0, or g = 0 and p 0 if there is contact. The complementarity condition g · p = 0, which enforces this alternative, was explicitly added by Moreau [86], probably inspired by the introduction of such a condition by Kuhn and Tucker [73] in optimization (independently of the thesis of Karush [71], which remained unnoticed until Kuhn and Tucker [73]). Therefore, although the three unilateral contact conditions are usually called Signorini's conditions, it seems fair to attribute them to Hertz, Signorini and Moreau in trio.…”

Section: Summary Of the Main Resultsmentioning

confidence: 99%

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Demoures

Gay–Balmaz

Raţiu

2016

Forum of Mathematics, Sigma

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This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In the presence of symmetry, a discrete version of the Noether theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

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Section: Summary Of the Main Resultsmentioning

confidence: 99%

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Demoures

Gay–Balmaz

Raţiu

2016

Forum of Mathematics, Sigma

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“…And, last but not least, the solutions dominate all others. The latter advantage can be fortified by evaluating the well known optimality criteria given by the Karush-Kuhn-Tucker (KKT) conditions (Kuhn and Tucker 1951). Such that the KKT can be plotted for all derived solutions p of Figs.…”

Section: Contrasting the Gradient-based Multi-criteria Optimization Wmentioning

confidence: 99%

Multi-criteria optimization of an aircraft propeller considering manufacturing

Schatz¹,

Hermanutz²,

Baier³

2016

Struct Multidisc Optim

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Carbon-fiber-reinforced polymers are gaining ground; high mass-specific stiffness and strength properties in fiber direction are commonly identified as reasons. Nonetheless, there are great challenges unleashing the entire light-weight potential. For instance, the multitude of parameters (e.g. fiber orientations), also being linked with each other and having huge influence on, not only structural mechanics, but also onto effort in terms of manufacturing. Moreover, these parameters ideally need to be determined, such that mass and costs are minimal, while all structural and technical requirements are fulfilled. The challenge of considering manufacturing aspects along with structural mechanics are mainly addressed in this paper. It is outlining an approach for modeling manufacturing effort via expert knowledge and how to actually consider this model in a multi-criteria optimization framework. In addition, it will be shown how to incorporate these knowledge-based models into an efficient structural design optimization. For this sake, a braided propeller structure is optimized.

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“…The high capacity and high speed computers have made it possible to develop new methods and to apply the optimum calculation for different problems. A method was developed by Kuhn and Tucker for solving the problem (1) with inequality constraints in 1951 [8]. The method was widely used from the 60-es.…”

Section: Nonlinear Programingmentioning

confidence: 99%

Mathematical optimization and engineering applications

Kulcsár¹,

Timár²

2016

Math. Model. Comput.

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New challenges raised almost in all segments of the global economy in the last two decades. The energy-, finance-, environmental crises, as well as the intensive use of natural resources require new design methods in the engineering activity, for both the product design and the manufacturing process planning. An effecting tool to be competitive is to apply the optimum design methods to the engineering tasks. This paper reviews the different methods of mathematical optimization, which are widely used for solving engineering problems, and describes the application of this method for two different cases. The first sample shows how to find the optimal dimensions of the welded, box type frame of a freight bogie, which will minimize the manufacturing cost of the structure and will satisfy several restrictions, e.g. mechanical stress limit, dimensional constraints, buckling, and fatigue conditions. The other case is to find the optimal geometric dimensions of a pipe insulation system which will result in a minimum investment and operating cost, when the heat loss and the outside surface temperature are restricted.

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Nonlinear Programming

Cited by 305 publications

References 0 publications

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Multi-criteria optimization of an aircraft propeller considering manufacturing

Mathematical optimization and engineering applications

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