On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals
In this paper, we introduce a new class of multi-dimensional robust optimization problems (named <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P...
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MDPI AG
2021-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/15/1790 |
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| author | Savin Treanţă Koushik Das |
| author_facet | Savin Treanţă Koushik Das |
| author_sort | Savin Treanţă |
| collection | DOAJ |
| description | In this paper, we introduce a new class of multi-dimensional robust optimization problems (named <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we define an auxiliary (modified) class of robust control problems (named <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>b</mi><mo stretchy="false">¯</mo></mover><mo>,</mo><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow></msub></semantics></math></inline-formula>), which is much easier to study, and provide some characterization results of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>b</mi><mo stretchy="false">¯</mo></mover><mo>,</mo><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow></msub></semantics></math></inline-formula> by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>b</mi><mo stretchy="false">¯</mo></mover><mo>,</mo><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow></msub></semantics></math></inline-formula>. For this aim, we consider path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) Lagrange 1-form. |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T09:12:37Z |
| publishDate | 2021-07-01 |
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| spelling | doaj.art-f6a85bcd0565484c8aba900b1458467f2023-11-22T05:56:44ZengMDPI AGMathematics2227-73902021-07-01915179010.3390/math9151790On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral FunctionalsSavin Treanţă0Koushik Das1Department of Applied Mathematics, University Politehnica of Bucharest, 060042 Bucharest, RomaniaDepartment of Mathematics, Taki Government College, Taki 743429, IndiaIn this paper, we introduce a new class of multi-dimensional robust optimization problems (named <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we define an auxiliary (modified) class of robust control problems (named <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>b</mi><mo stretchy="false">¯</mo></mover><mo>,</mo><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow></msub></semantics></math></inline-formula>), which is much easier to study, and provide some characterization results of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>b</mi><mo stretchy="false">¯</mo></mover><mo>,</mo><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow></msub></semantics></math></inline-formula> by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>b</mi><mo stretchy="false">¯</mo></mover><mo>,</mo><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow></msub></semantics></math></inline-formula>. For this aim, we consider path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) Lagrange 1-form.https://www.mdpi.com/2227-7390/9/15/1790Lagrange 1-formsecond-order Lagrangiannormal weak robust optimal solutionmodified objective function methodrobust saddle-point |
| spellingShingle | Savin Treanţă Koushik Das On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals Mathematics Lagrange 1-form second-order Lagrangian normal weak robust optimal solution modified objective function method robust saddle-point |
| title | On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals |
| title_full | On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals |
| title_fullStr | On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals |
| title_full_unstemmed | On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals |
| title_short | On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals |
| title_sort | on robust saddle point criterion in optimization problems with curvilinear integral functionals |
| topic | Lagrange 1-form second-order Lagrangian normal weak robust optimal solution modified objective function method robust saddle-point |
| url | https://www.mdpi.com/2227-7390/9/15/1790 |
| work_keys_str_mv | AT savintreanta onrobustsaddlepointcriterioninoptimizationproblemswithcurvilinearintegralfunctionals AT koushikdas onrobustsaddlepointcriterioninoptimizationproblemswithcurvilinearintegralfunctionals |