On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals

• Main Authors: Savin Treanţă, Koushik Das
• Format: Article
• Language: English
• Published: MDPI AG 2021-07-01
• Series: Mathematics
• Subjects: Lagrange 1-form; second-order Lagrangian; normal weak robust optimal solution; modified objective function method; robust saddle-point
• Online Access: https://www.mdpi.com/2227-7390/9/15/1790

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.