In this work we present some exactness conditions for the Shor relaxation of diagonal (or, more generally, diagonalizable) QCQPs, which extend introduced in different recent papers about same topic. It is shown that equivalent to two convex quadratic relaxations. Then, sufficient relaxations are derived from their KKT systems. will be that, cases, by derivation previous literature, can viewed a...

By considering the epigraphs of conjugate functions, we extend the Fenchel duality, applicable to a (possibly infinite) family of proper lower semicontinuous convex functions on a Banach space. Applications are given in providing fuzzy KKT conditions for semi-infinite programming.

Two interior-point algorithms are proposed and analyzed, for the (local) solution of (possibly) indefinite quadratic programming problems. They are of the Newton-KKT variety in that (much like in the case of primal-dual algorithms for linear programming) search directions for the “primal” variables and the Karush-Kuhn-Tucker (KKT) multiplier estimates are components of the Newton (or quasi-Newt...

The global convergence of ADMM for convex problems was given by He and Yuan in [5] under the variation inequality framework. However, since our optimization problem is non-convex, the convergence analysis for ADMM needs additional conditions. In non-convex optimization, convergence to a stationary point (local minimum) is the best convergence property that we can hope for. By imposing some cond...

We consider optimality systems of Karush-Kuhn-Tucker (KKT) type, which arise, for example, as primal-dual conditions characterizing solutions of optimization problems or variational inequalities. In particular, we discuss error bounds and Newton-type methods for such systems. An exhaustive comparison of various regularity conditions which arise in this context is given. We obtain a new error bo...

We consider a linear complementarity problem (LCP) arisen from the Arrow-Debreu-Leontief competitive economy equilibrium where the LCP coefficient matrix is symmetric. We prove that the decision problem, to decide whether or not there exists a complementary solution, is NP-complete. Under certain conditions, an LCP solution is guaranteed to exist and we present a fully polynomial-time approxima...

In this paper, we consider using the neural networks to efficiently solve the second-order cone constrained variational inequality (SOCCVI) problem. More specifically, two kinds of neural networks are proposed to deal with the Karush-Kuhn-Tucker (KKT) conditions of the SOCCVI problem. The first neural network uses the FischerBurmeister (FB) function to achieve an unconstrained minimization whic...

The Fisher discriminant analysis (FDA) is a common technique for binary classification. A parametrized extension, which we call the extended FDA, has been introduced from the viewpoint of robust optimization. In this work, we first give a new probabilistic interpretation of the extended FDA. We then develop algorithms for solving an optimization problem that arises from the extended FDA: comput...

In this report we describe how continuation methods can be used for the numerical treatment of multi-objective optimization problems (MOPs): starting with a given Karush-Kuhn-Tucker point (KKTpoint) x̃ of an MOP, these techniques can be applied to detect further KKT-points in the neighborhood of x̃. In the next step, again further points are computed starting with these new-found KKT-points, and ...