constraint qualifications

Two New Weak Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and Applications

  • Alberto Ramos
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags ,

    We introduce two new weaker Constraint Qualifications (CQs) for Mathematical Programs with Equilibrium (or Complementarity) Constraints, MPEC for short. One of them is a tailored version of the Constant Rank of Subspace Component (CRSC) and the other is a relaxed version of the MPEC-No Nonzero Abnormal Multiplier Constraint Qualification (MPEC-NNAMCQ). Both incorporate the exact set … Read more

    A second-order optimality condition with first- and second-order complementarity associated with global convergence of algorithms

  • Gabriel Haeser
    • Categories Constrained Nonlinear Optimization Tags , , ,

    We develop a new notion of second-order complementarity with respect to the tangent subspace related to second-order necessary optimality conditions by the introduction of so-called tangent multipliers. We prove that around a local minimizer, a second-order stationarity residual can be driven to zero while controlling the growth of Lagrange multipliers and tangent multipliers, which gives … Read more

    On a conjecture in second-order optimality conditions

  • Roger Behling
  • Gabriel Haeser
  • Alberto Ramos
  • Daiana O. Santos
    • Categories Constrained Nonlinear Optimization Tags , , ,

    In this paper we deal with optimality conditions that can be verified by a nonlinear optimization algorithm, where only a single Lagrange multiplier is avaliable. In particular, we deal with a conjecture formulated in [R. Andreani, J.M. Martinez, M.L. Schuverdt, “On second-order optimality conditions for nonlinear programming”, Optimization, 56:529–542, 2007], which states that whenever a … Read more

    Strict Constraint Qualifications and Sequential Optimality Conditions for Constrained Optimization

  • Roberto Andreani
  • José Mario Martínez
  • Alberto Ramos
  • Paulo J. S. Silva
    • Categories Constrained Nonlinear Optimization Tags , , , , , ,

    Sequential optimality conditions for constrained optimization are necessarily satisfied by local minimizers, independently of the fulfillment of constraint qualifications. These conditions support the employment of different stopping criteria for practical optimization algorithms. On the other hand, when an appropriate strict constraint qualification associated with some sequential optimality condition holds at a point that satisfies the … Read more

    A second-order sequential optimality condition associated to the convergence of optimization algorithms

  • Roberto Andreani
  • Gabriel Haeser
  • Alberto Ramos
  • Paulo J. S. Silva
    • Categories Constrained Nonlinear Optimization Tags , ,

    Sequential optimality conditions have recently played an important role on the analysis of the global convergence of optimization algorithms towards first-order stationary points and justifying their stopping criteria. In this paper we introduce the first sequential optimality condition that takes into account second-order information. We also present a companion constraint qualification that is less stringent … Read more

    First order optimality conditions for mathematical programs with second-order cone complementarity constraints

  • Jane J. Ye
  • Jinchuan Zhou
    • Categories Complementarity and Variational Inequalities, Nonsmooth Optimization, Second-Order Cone Programming Tags , , , , ,

    In this paper we consider a mathematical program with second-order cone complementarity constraints (SOCMPCC). The SOCMPCC generalizes the mathematical program with complementarity constraints (MPCC) in replacing the set of nonnegative reals by a second-order cone. We show that if the SOCMPCC is considered as an optimization problem with convex cone constraints, then Robinson’s constraint qualification … Read more

    A cone-continuity constraint qualification and algorithmic consequences

  • Roberto Andreani
  • José Mario Martínez
  • Alberto Ramos
  • Paulo J. S. Silva
    • Categories Constrained Nonlinear Optimization Tags , , , ,

    Every local minimizer of a smooth constrained optimization problem satisfies the sequential Approximate Karush-Kuhn-Tucker (AKKT) condition. This optimality condition is used to define the stopping criteria of many practical nonlinear programming algorithms. It is natural to ask for conditions on the constraints under which AKKT implies KKT. These conditions will be called Strict Constraint Qualifications … Read more

    Constraint Qualification Failure in Action

  • Hassan L. Hijazi
  • Leo Liberti
    • Categories (Mixed) Integer Nonlinear Programming, Second-Order Cone Programming Tags , , , ,

    This note presents a theoretical analysis of disjunctive constraints featuring unbounded variables. In this framework, classical modeling techniques, including big-M approaches, are not applicable. We introduce a lifted second-order cone formulation of such on/off constraints and discuss related constraint qualification issues. A solution is proposed to avoid solvers’ failure. Citation H. L. Hijazi and L.Liberti … Read more

    The Slater Conundrum: Duality and Pricing in Infinite Dimensional Optimization

  • Kipp Martin
  • Christopher Thomas Ryan
  • Matthew Stern
    • Categories Infinite Dimensional Optimization Tags , , , ,

    Duality theory is pervasive in finite dimensional optimization. There is growing interest in solving infinite-dimensional optimization problems and hence a corresponding interest in duality theory in infinite dimensions. Unfortunately, many of the intuitions and interpretations common to finite dimensions do not extend to infinite dimensions. In finite dimensions, a dual solution is represented by a … Read more

    On local convergence of the method of alternating projections

  • Dominikus Noll
  • Aude Rondepierre
    • Categories Convex and Nonsmooth Optimization Tags , , , , , ,

    The method of alternating projections is a classical tool to solve feasibility problems. Here we prove local convergence of alternating projections between subanalytic sets A,B under a mild regularity hypothesis on one of the sets. We show that the speed of convergence is O$(k^{-\rho})$ for some $\rho\in(0,\infty)$. Citation Université de Toulouse, Institut de Mathématiques, december … Read more