Semi-infinite programming

<h2 id="mathematical-formulation-of-the-problem">Mathematical formulation of the problem</h2>
<p>The problem can be stated simply as:
</p>

min
          
            x
            ∈
            X
          
        
        
        
        f
        (
        x
        )
      
    
    {\displaystyle \min _{x\in X}\;\;f(x)}

subject to: 
        
      
    
    {\displaystyle {\text{subject to: }}}

g
        (
        x
        ,
        y
        )
        ≤
        0
        ,
        
        
        ∀
        y
        ∈
        Y
      
    
    {\displaystyle g(x,y)\leq 0,\;\;\forall y\in Y}

<p>where
</p>

f
        :
        
          R
          
            n
          
        
        →
        R
      
    
    {\displaystyle f:R^{n}\to R}

g
        :
        
          R
          
            n
          
        
        ×
        
          R
          
            m
          
        
        →
        R
      
    
    {\displaystyle g:R^{n}\times R^{m}\to R}

X
        ⊆
        
          R
          
            n
          
        
      
    
    {\displaystyle X\subseteq R^{n}}

Y
        ⊆
        
          R
          
            m
          
        
        .
      
    
    {\displaystyle Y\subseteq R^{m}.}

<p>SIP can be seen as a special case of <a href="/facts/Bilevel_program/s1k2ALp3">bilevel programs</a> in which the lower-level variables do not participate in the objective function.
</p>
<h2 id="methods-for-solving-the-problem">Methods for solving the problem</h2>

<p>In the meantime, see external links below for a complete tutorial.
</p>
<h2 id="examples">Examples</h2>

<p>In the meantime, see external links below for a complete tutorial.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Optimization_(mathematics)/oRn8Iv5I">Optimization</a></li>
<li><a href="/facts/Generalized_semi-infinite_programming/8le1yTnb">Generalized semi-infinite programming (GSIP)</a></li></ul>

<ul><li>Anderson, Edward J.; Nash, Peter (1987). <i>Linear Programming in Infinite-Dimensional Spaces</i>. Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-91250-6. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/15053031">15053031</a>.</li>
<li>Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4, 7.4.4 Semi-infinite programming". <i>Perturbation analysis of optimization problems</i>. Springer Series in Operations Research. Springer. pp. 496–526, 581. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-98705-7. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1756264">1756264</a>.</li>
<li>Goberna, M.A.; López, M.A. (1998). <i>Linear Semi-Infinite Optimization</i>. Wiley.</li>
<li>Goberna, M.A.; López, M.A. (2014). <a href="https://link.springer.com/book/10.1007/978-1-4899-8044-1"><i>Post-Optimal Analysis in Linear Semi-Infinite Optimization</i></a>. SpringerBriefs in Optimization. Springer. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4899-8044-1">10.1007/978-1-4899-8044-1</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4899-8044-1.</li>
<li>Hettich, R.; Kortanek, K.O. (1993). "Semi-infinite programming: Theory, methods, and applications". <i>SIAM Review</i>. 35 (3): 380–429. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1137%2F1035089">10.1137/1035089</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2132425">2132425</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1234637">1234637</a>.</li>
<li>Luenberger, David G. (1997). <i>Optimization by Vector Space Methods</i>. Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-18117-X. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/52405793">52405793</a>.</li>
<li>Reemtsen and, Rembert; Rückmann, Jan-J., eds. (1998). <a href="https://link.springer.com/book/10.1007/978-1-4757-2868-2"><i>Semi-Infinite Programming</i></a>. Nonconvex Optimization and Its Applications. Vol. 25. Springer. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4757-2868-2">10.1007/978-1-4757-2868-2</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4757-2868-2.</li>
<li>Guerra Vázquez, F.; Rückmann, J.-J.; Stein, O.; Still, G. (1 August 2008). <a href="http://www.sciencedirect.com/science/article/pii/S0377042707000982">"Generalized semi-infinite programming: A tutorial"</a>. <i>Journal of Computational and Applied Mathematics</i>. 217 (2): 394–419. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2008JCoAM.217..394G">2008JCoAM.217..394G</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.cam.2007.02.012">10.1016/j.cam.2007.02.012</a>.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://glossary.informs.org/ver2/mpgwiki/index.php?title=Semi-infinite_program">Description of semi-infinite programming from INFORMS (Institute for Operations Research and Management Science)</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Bonnans & Shapiro 2000, pp. 496–526, 581
Goberna & López 1998
Hettich & Kortanek 1993, pp. 380–429 - Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4, 7.4.4 Semi-infinite programming". Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer. pp. 496–526, 581. ISBN 978-0-387-98705-7. MR 1756264. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1756264" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1756264</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Semi-infinite programming open-in-new

Semi-infinite programming