An Effective Algorithm for Quadratic Optimization with Non-Convex Inhomogeneous Quadratic Constraints

We derive computationally tractable formulations of the robust counterparts of convex quadratic and conic quadratic constraints that are concave in matrix-valued uncertain parameters. We do this for a broad range of uncertainty sets. Our results provide extensions to known results from the literature. We also consider hard quadratic constraints: those that are convex in uncertain matrix-valued parameters. For the robust counterpart of such constraints, we derive inner and outer tractable approximations. As an application, we show how to construct a natural uncertainty set based on a statistical confidence set around a sample mean vector and covariance matrix and use this to provide a tractable reformulation of the robust counterpart of an uncertain portfolio optimization problem. We also apply the results of this paper to norm approximation problems. Summary of Contribution: This paper develops new theoretical results and algorithms that extend the scope of a robust quadratic optimization problem. More specifically, we derive computationally tractable formulations of the robust counterparts of convex quadratic and conic quadratic constraints that are concave in matrix-valued uncertain parameters. We also consider hard quadratic constraints: those that are convex in uncertain matrix-valued parameters. For the robust counterpart of such constraints, we derive inner and outer tractable approximations.

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An effective algorithm for globally solving quadratic programs using parametric linearization technique

Open Mathematics ◽

10.1515/math-2018-0108 ◽

2018 ◽

Vol 16 (1) ◽

pp. 1300-1312

Author(s):

Shuai Tang ◽

Yuzhen Chen ◽

Yunrui Guo

Keyword(s):

Linear Programming ◽

Optimal Solution ◽

Route Optimization ◽

Linear Programming Relaxation ◽

Effective Algorithm ◽

Quadratic Programs ◽

Parametric Linear Programming ◽

Quadratic Constraints ◽

Linearization Technique ◽

Relaxation Problem

AbstractIn this paper, we present an effective algorithm for globally solving quadratic programs with quadratic constraints, which has wide application in engineering design, engineering optimization, route optimization, etc. By utilizing new parametric linearization technique, we can derive the parametric linear programming relaxation problem of the quadratic programs with quadratic constraints. To improve the computational speed of the proposed algorithm, some interval reduction operations are used to compress the investigated interval. By subsequently partitioning the initial box and solving a sequence of parametric linear programming relaxation problems the proposed algorithm is convergent to the global optimal solution of the initial problem. Finally, compared with some known algorithms, numerical experimental results demonstrate that the proposed algorithm has higher computational efficiency.

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A copositive Farkas lemma and minimally exact conic relaxations for robust quadratic optimization with binary and quadratic constraints

Operations Research Letters ◽

10.1016/j.orl.2019.09.013 ◽

2019 ◽

Vol 47 (6) ◽

pp. 530-536 ◽

Cited By ~ 1

Author(s):

N.H. Chieu ◽

T.D. Chuong ◽

V. Jeyakumar ◽

G. Li

Keyword(s):

Quadratic Optimization ◽

Quadratic Constraints ◽

Farkas Lemma

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Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints

Journal of Global Optimization ◽

10.1007/s10898-010-9545-5 ◽

2010 ◽

Vol 49 (2) ◽

pp. 293-311 ◽

Cited By ~ 13

Author(s):

Xinzhen Zhang ◽

Chen Ling ◽

Liqun Qi

Keyword(s):

Optimization Problems ◽

Quadratic Optimization ◽

Semidefinite Relaxation ◽

Quadratic Constraints

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SDO and LDO relaxation approaches to complex fractional quadratic optimization

RAIRO - Operations Research ◽

10.1051/ro/2020090 ◽

2020 ◽

Author(s):

ali ashrafi ◽

Arezu Zare

Keyword(s):

Optimization Problem ◽

Original Problem ◽

Quadratic Optimization ◽

Parametric Quadratic Programming ◽

Quadratic Constraints ◽

Parametric Problem ◽

Quadratic Optimization Problem ◽

Lagrangian Dual ◽

Fractional Quadratic Optimization ◽

Generalized Newton

This paper examines a complex fractional quadratic optimization problem subject to two quadratic constraints. The original problem is transformed into a parametric quadratic programming problem by the well-known classical Dinkelbach method. Then a semidefinite and Lagrangian dual optimization approaches are presented to solve the nonconvex parametric problem at each iteration of the bisection and generalized Newton algorithms. Finally, the numerical results demonstrate the effectiveness of the proposed approaches.

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