An interior-point method for nonlinear optimization problems with locatable and separable nonsmoothness

Schmidt, Martin

doi:10.1007/s13675-015-0039-6

Cited by 14 publications

(4 citation statements)

References 29 publications

(32 reference statements)

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“…Recently, Hahn et al [81] construct a finite volume scheme for the PDE constraints and present heuristics to solve the resulting MINLP. Moreover, extended nonlinear models of mathematical programs with equilibrium constraints (MPEC) that allow for switching of discrete states within continuous models are investigated; see Baumrucker and Biegler [10], Schmidt [138,139], and Schmidt et al [143,144].…”

Section: Literature Survey: Solving Gas Transport Optimization Problemsmentioning

confidence: 99%

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

Sirvent

2020

Operations Research Proceedings

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List of Figures List of Tables 1.3 Scientific Contribution and Results challenge is to find formulations and corresponding algorithms that are able to tackle transient gas transport optimization problems. Note that the first challenge seeks for general algorithms that are then used for stationary gas transport optimization, whereas the second challenge is tailored to the application of transient gas transport optimization. Our roadmap to cope with the challenges is inspired by two guidelines. We want to decompose our problems and we want to use mixed-integer linear programs. The simple reason for the first point is that decomposing problems into smaller ones has a successful history and is common sense for solving big problems. Second, mixed-integer linear programs underwent an enormous amount of research in the last decades. Consequently, fast, stable, robust, and reliable solvers as Gurobi, Cplex, or Scip are available and we use these solvers, in our case Gurobi, as a workhorse; see Gurobi [80], Cplex [36], and Maher et al. [110], respectively. 1.3 Scientific Contribution and Results For the first challenge, we consider the case where the constraint functions in (1.1c) are not given analytically. We build up a new hierarchy of assumptions that restricts the knowledge about the nonlinear functions. Depending on the assumptions, we develop three new global decomposition algorithms that decompose the difficult constraints (1.1c), i.e., we rely on our first guideline. Specifically, we build upon relaxation strategies that are motivated by developments in the field of (mixed-integer) (non)linear programming and Lipschitz optimization. As a result, mixed-integer linear programs according to our second guideline are obtained. We prove correct and finite terminations and clarify the limitations of our approaches. Note that the presented framework is a general approach for problems in the form of (1.1). In particular, we consider stationary gas transport optimization, i.e., Problem (1.1) with the Euler equations in the form of ODEs in (1.1c). Moreover, we show promising numerical results for the real-world gas network of Greece. Additionally, we consider the second challenge of transient gas transport optimization, i.e., Problem (1.1) with PDEs (1.1c). We use an instantaneous control algorithm and adapt it to the Euler equations for the first time. Specifically, we decompose the problem into single time steps and present new discretization techniques that ensure the mixed-integer linear program property. In other words, we build upon our guidelines again. Finally, we

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Section: Literature Survey: Solving Gas Transport Optimization Problemsmentioning

confidence: 99%

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

Sirvent

2020

Operations Research Proceedings

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“…Step 3: The bi-objective subproblem P µ NEEDP is transformed into a set of single-objective subproblems using the -constraint method. For both methods, the objective function of the subproblems are smoothed by the smoothing method and the subproblems are solved by the interior point barrier method [15].…”

Section: Applicationmentioning

confidence: 99%

Solving Nonsmooth Bi-Objective Environmental andEconomic Dispatch Problem using Smoothing Techniques

Tifroute,

Lahmdani,

Bouzahir

2020

Preprint

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The Environmental and Economic Dispatch problem (EEDP) [14,13] is a nonlinear Multi-objective Optimization Problem (MOP) which simultaneously satisfies multiple contradictory criteria, and it's a nonsmooth problem when valvepoint effects, multi-fuel effects and prohibited operating zones have been considered. It is an important optimization task in fossil fuel fired power plant operation for allocating generation among the committed units such that fuel cost and pollution (emission level) are optimized simultaneously while satisfying all operational constraints. In this paper, we use smoothing functions with the gradient consistency property to approximate the nonsmooth multi-objective Optimization problem. Our approach is based on the smoothing method. In fact, we explain the convergence analysis of smoothing method by using approximate Karush-Kuhn-Tucker condition, which is necessary for a point to be a local weak efficient solution and is also sufficient under convexity assumptions. Finally, we give an application of our approach for solving the bi-objective EEDP.

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“…Zhang et al [13] presented an interior-point method-based program combined with the homogenization theory on micromechanics to analyze shakedown behavior at the macro-scale. Schmidt [14] presented an interior-point method for nonlinear optimization problems with locatable and separable no-smoothness. Sakineh [15] presented novel interior-point algorithms for solving nonlinear convex optimization problems.…”

Section: Introductionmentioning

confidence: 99%

Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections

et al. 2021

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In this paper, a penalty method is used together with a barrier method to transform a constrained nonlinear programming problem into an unconstrained nonlinear programming problem. In the proposed approach, Newton’s method is applied to the barrier Karush–Kuhn–Tucker conditions. To ensure global convergence from any starting point, a trust-region globalization strategy is used. A global convergence theory of the penalty–barrier trust-region (PBTR) algorithm is studied under four standard assumptions. The PBTR has new features; it is simpler, has rapid convergerce, and is easy to implement. Numerical simulation was performed on some benchmark problems. The proposed algorithm was implemented to find the optimal design of a canal section for minimum water loss for a triangle cross-section application. The results are promising when compared with well-known algorithms.

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An interior-point method for nonlinear optimization problems with locatable and separable nonsmoothness

Cited by 14 publications

References 29 publications

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

Solving Nonsmooth Bi-Objective Environmental andEconomic Dispatch Problem using Smoothing Techniques

Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections

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