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<datestamp>2017-12-21</datestamp>
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<publisher>HAL CCSD</publisher>
<title lang=en>Subdifferential estimate of the directional derivative, optimality criterion and separation principles</title>
<creator>Jules, Florence</creator>
<creator>Lassonde, Marc</creator>
<contributor>Laboratoire de Mathématiques Informatique et Applications (LAMIA) ; Université des Antilles et de la Guyane (UAG)</contributor>
<description>International audience</description>
<source>ISSN: 0233-1934</source>
<source>Optimization</source>
<publisher>Taylor & Francis</publisher>
<identifier>hal-00694601</identifier>
<identifier>https://hal.archives-ouvertes.fr/hal-00694601</identifier>
<source>https://hal.archives-ouvertes.fr/hal-00694601</source>
<source>Optimization, Taylor & Francis, 2013, 62 (9), pp.1267-1288. 〈10.1080/02331934.2011.645034〉</source>
<identifier>DOI : 10.1080/02331934.2011.645034</identifier>
<relation>info:eu-repo/semantics/altIdentifier/doi/10.1080/02331934.2011.645034</relation>
<language>en</language>
<subject lang=en>-subdifferential</subject>
<subject lang=en>lower semicontinuity</subject>
<subject lang=en>subdifferential</subject>
<subject lang=en>"-subdifferential</subject>
<subject lang=en>directional derivative</subject>
<subject lang=en>optimality criteria</subject>
<subject lang=en>maximal monotonicity</subject>
<subject lang=en>mean value inequality</subject>
<subject lang=en>separation principle</subject>
<subject>[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]</subject>
<type>info:eu-repo/semantics/article</type>
<type>Journal articles</type>
<description lang=en>We provide an inequality relating the radial directional derivative and the subdifferential of proper lower semicontinuous functions, which extends the known formula for convex functions. We show that this property is equivalent to other subdifferential properties of Banach spaces, such as controlled dense subdifferentiability, optimality criterion, mean value inequality and separation principles. As an application, we obtain a first-order sufficient condition for optimality, which extends the known condition for differentiable functions in finite-dimensional spaces and which amounts to the maximal monotonicity of the subdifferential for convex lower semicontinuous functions. Finally, we establish a formula describing the subdifferential of the sum of a convex lower semicontinuous function with a convex inf-compact function in terms of the sum of their approximate epsilon-subdifferentials. Such a formula directly leads to the known formula relating the directional derivative of a convex lower semicontinuous function to its approximate epsilon-subdifferential.</description>
<date>2013-08-01</date>
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