Optimization Problem Coupled with Differential Equations: A Numerical Algorithm Mixing an Interior-Point Method and Event Detection

Caboussat, Alexandre; Landry, Chantal; Rappaz, Jacques

doi:10.1007/s10957-010-9714-1

Cited by 4 publications

(6 citation statements)

References 17 publications

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“…Generally speaking, the vertices of ∆ 1 correspond to pure solutions, and any point between 0 and 1 corresponds to a mixing of these two components. Figure 3 illustrates the computation of the convex envelope for the pinic acid-hexacosanol system, by using a dynamic approach detailed in [9,19,20]. The supporting tangent planes are illustrated at various points d. The union of these cuts determines the convex hull of the Gibbs free energy function.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The total computational cost is 20 s. when using a grid of Finally, Figure 5 is extracted from [19] and visualizes the computation of the convex envelope for a four components chemical system (composed of water, pinic acid, 1-hexacosanol and npropanol) illustrated in ∆ 3 . The value of the convex envelope is computed for a sequence of points located on a trajectory in ∆ 3 modeled in [1,9,20]. The trajectory is projected on the sides of ∆ 3 for visualization purposes.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Initialization of (y Step 3. Compute a step-length τ to ensure that y Numerical results are given in Section 7., for various chemical systems, following [4,9,19]. In the next section, we first present an alternative approach based on variational arguments, for the computation of the convex envelopes.…”

Section: Primal-dual Interior-point Methodsmentioning

confidence: 99%

“…Besides the macroscopic operators (advection and diffusion), the microscopic processes involve phase transformation and chemical reactions [24,25,26]. The two main processes involving the particle phase are the thermodynamic equilibrium [4,6] inside the particle and the mass transfer between the gas phase and the particle phase [1,9,20].…”

Section: Microscopic Modeling Of Aerosol Particlesmentioning

confidence: 99%

“…Numerical methods for the solution of (3.1) (3.2) have been proposed in [9,20,32]. In the remainder of this article, we focus more precisely on this global optimization problem (3.3) and propose two solution methods for the computation of the thermodynamic equilibrium inside each organic particle.…”

Section: Microscopic Modeling Of Aerosol Particlesmentioning

confidence: 99%

See 4 more Smart Citations

Mathematical Modeling of Atmospheric Flow and Computation of Convex Envelopes

Caboussat

2011

Math. Model. Nat. Phenom.

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Abstract. Atmospheric flow equations govern the time evolution of chemical concentrations in the atmosphere. When considering gas and particle phases, the underlying partial differential equations involve advection and diffusion operators, coagulation effects, and evaporation and condensation phenomena between the aerosol particles and the gas phase. Operator splitting techniques are generally used in global air quality models. When considering organic aerosol particles, the modeling of the thermodynamic equilibrium of each particle leads to the determination of the convex envelope of the energy function. Two strategies are proposed to address the computation of convex envelopes. The first one is based on a primal-dual interior-point method, while the second one relies on a variational formulation, an appropriate augmented Lagrangian, an Uzawa iterative algorithm, and finite element techniques. Numerical experiments are presented for chemical systems of atmospheric interest, in order to compute convex envelopes in various space dimensions.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Numerical Resultsmentioning

confidence: 99%

Section: Primal-dual Interior-point Methodsmentioning

confidence: 99%

Section: Microscopic Modeling Of Aerosol Particlesmentioning

confidence: 99%

Section: Microscopic Modeling Of Aerosol Particlesmentioning

confidence: 99%

See 3 more Smart Citations

Mathematical Modeling of Atmospheric Flow and Computation of Convex Envelopes

Caboussat

2011

Math. Model. Nat. Phenom.

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Analysis of the local well‐posedness of optimization‐constrained differential equations by local optimality conditions

et al. 2020

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Optimization‐constrained differential equations (OCDE) are a class of mathematical problems where differential equations are constrained by an embedded algebraic optimization problem. We analyze the well‐posedness of the local solutions of OCDE based on local optimality. By assuming linear independence constraint qualification and applying the Karush‐Kuhn‐Tucker optimality conditions, an OCDE is transformed into a complementarity system (CS). Under second‐order sufficient condition we show that (a) if strict complementary condition (SCC) holds, the local solution of OCDE is well‐posed, which corresponds to a mode of the derived CS; (b) at points where SCC is violated, a local solution of OCDE exists by sequentially connecting the local solutions of two selected modes of the derived CS. We propose an event‐based algorithm to numerically solve OCDE. We illustrate the approach and algorithm for microbial cultivation, single flash unit and contrived numerical examples.

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Dynamic flux balance analysis with nonlinear objective function

et al. 2017

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Dynamic flux balance analysis (DFBA) extends flux balance analysis and enables the combined simulation of both intracellular and extracellular environments of microbial cultivation processes. A DFBA model contains two coupled parts, a dynamic part at the upper level (extracellular environment) and an optimization part at the lower level (intracellular environment). Both parts are coupled through substrate uptake and product secretion rates. This work proposes a Karush-Kuhn-Tucker condition based solution approach for DFBA models, which have a nonlinear objective function in the lower-level part. To solve this class of DFBA models an extreme-ray-based reformulation is proposed to ensure certain regularity of the lower-level optimization problem. The method is introduced by utilizing two simple example networks and then applied to a realistic model of central carbon metabolism of wild-type Corynebacterium glutamicum.

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Optimization Problem Coupled with Differential Equations: A Numerical Algorithm Mixing an Interior-Point Method and Event Detection

Cited by 4 publications

References 17 publications

Mathematical Modeling of Atmospheric Flow and Computation of Convex Envelopes

Mathematical Modeling of Atmospheric Flow and Computation of Convex Envelopes

Analysis of the local well‐posedness of optimization‐constrained differential equations by local optimality conditions

Dynamic flux balance analysis with nonlinear objective function

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