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Convergent Algorithms for a Class of Convex Semi-infinite Programs

“…Moreover, the finite single-level formulation we obtain with this approach is convex, contrary to the pre-existing methods in [10,25]. When g(x, y) is not convex in y, we still have convergence guarantees.…”

“…Lower-level duality has been already used in [10,25], leveraging on approaches from robust optimization [5]. However, contrary to what is proposed in this paper, the existing dual approaches lead to non-convex problems, and do not use SDP.…”

“…(PSD), the obtained formulation is not a restriction, but a reformulation of (SIP). The dualization technique has been used in the SIP literature in [10,25], and inspired by approaches from robust optimization [4,]. However, SDP together with Lagrangian duality has never been used to dualize a quadratic programming lower-level problem to the best of our knowledge.…”

“…In [10], several strategies are used to reformulate generalized SIP problems into non-convex finite minimization problems by exploiting Wolfe duality for the convex lower-level problems. In [25], the authors tackle generalized SIP problems where the convex quadratic lower-level problem has a fixed Hessian matrix Q, which does not depend on the variable x. Instead, in the present paper, we consider standard SIP problems with a linear function Q(x) for the lower-level Hessian (as stated in Assumption 3).…”

Mixed integer semi-infinite programming problem

“…近年来, 半无限规划 (semi-infinite programming, SIP) 问题被广泛研究 (参见文献 [1][2][3][4][5][6][7][8]). 绝大多 数文献讨论的是只有连续变量的情形, 求解这种 SIP 的常用方法是将其转化为标准的线性/非线性规 划问题.…”