scholarly journals The Generalized Trust-Region Sub-Problem with Additional Linear Inequality Constraints—Two Convex Quadratic Relaxations and Strong Duality

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1369
Author(s):  
Temadher A. Almaadeed ◽  
Akram Taati ◽  
Maziar Salahi ◽  
Abdelouahed Hamdi
Keyword(s):  
Linear Inequality ◽  
Sufficient Conditions ◽  
Optimal Solution ◽  
Quadratic Function ◽  
Inequality Constraints ◽  
Fixed Number ◽  
Semidefinite Optimization ◽  
Global Optimality ◽  
Quadratic Constraints ◽  
Linear Inequality Constraints

In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different conditions, that are minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints. Then, we discuss cases where the CQRs return the optimal solution of the problem, revealing new conditions under which the underlying problem admits strong Lagrangian duality and enjoys exact semidefinite optimization relaxation. Finally, under the given sufficient conditions, we present necessary and sufficient conditions for global optimality of the problem and obtain a form of S-lemma for a system of two quadratic and a fixed number of linear inequalities.

Optimal Ship Forms for Minimum Total Resistance

1984 ◽  
Vol 28 (03) ◽  
pp. 163-172
Author(s):  
Chi-Chao Hsiung ◽  
Dong Shenyan
Keyword(s):  
Quadratic Form ◽  
Numerical Scheme ◽  
Linear Inequality ◽  
Optimal Solution ◽  
Inequality Constraints ◽  
Frictional Resistance ◽  
Wave Resistance ◽  
Total Resistance ◽  
Linear Inequality Constraints ◽  
Quadratic Programming Method

A numerical scheme is developed by using the "tent" function to compute the hull surface area and then the ship frictional resistance in a quadratic form in terms of ship offsets. Combining with the wave resistance, a total ship resistance formula is derived in a standard quadratic form. With a set of linear-inequality constraints, the optimal solution of ship offsets for minimum total resistance can be obtained by applying a quadratic programming method to the problem. Computations have been carried out for three conditions which have exactly the same constraints as those required to obtain the optimal forms for minimum wave resistance shown in an earlier work [1].3 The optimal forms found for minimum total resistance also have either bow or midship bulbs. The new optimal bulbs have a relatively small size, higher vertical slope to the baseline, and less curvature at the tip of the bulb.

scholarly journals Quadratic problems with two quadratic constraints: convex quadratic relaxation and strong Lagrangian duality

2020 ◽  
Author(s):  
Abdelouahed Hamdi ◽  
Akram Taati ◽  
Temadher A Almaadeed
Keyword(s):  
Sufficient Conditions ◽  
Optimal Solution ◽  
Lagrangian Duality ◽  
Global Optimality ◽  
Global Optimal Solution ◽  
Quadratic Constraints ◽  
Quadratic Minimization ◽  
Global Optimal ◽  
Necessary And Sufficient ◽  
Convex Quadratic Relaxation

In this paper,  we study  a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex.  We introduce two convex quadratic relaxations (CQRs) and discuss cases, where the problem is equivalent to exactly one of the CQRs. Particularly, we show that the global optimal  solution can be recovered from an optimal solution of the CQRs. Through this equivalence, we introduce new conditions under which the problem enjoys strong Lagrangian duality, generalizing  the recent  condition  in the literature.  Finally, under the new conditions,  we present  necessary and sufficient conditions for global optimality of the problem.

scholarly journals Mathematical model for determining rational machining conditions for flat grinding with the wheel periphery on machines with a rectangular table

2018 ◽  
Vol 224 ◽  
pp. 01112
Author(s):  
Dmitriy L. Skuratov ◽  
Dmitriy G. Fedorov ◽  
Dmitriy V. Evdokimov
Keyword(s):  
Mathematical Model ◽  
Objective Function ◽  
Linear Inequality ◽  
Inequality Constraints ◽  
Machining Time ◽  
Functional Parameters ◽  
Linear Inequality Constraints ◽  
Machining Quality ◽  
Machining Conditions ◽  
Grinding Operations

A mathematical model is presented for determining the rational machining conditions for flat grinding operations by the rim of a wheel on machines with a rectangular table consisting of a linear objective function and linear inequality constraints. As the objective function, the equation, determining the main machining time, was used. And constraints which are related to the functional parameters and parameters determining the machining quality and the kinematic capabilities of the machine were used as inequality constraints.

scholarly journals Linear programming with inequality constraints via entropic perturbation

1996 ◽  
Vol 19 (1) ◽  
pp. 177-184 ◽  
Author(s):  
H.-S. Jacob Tsao ◽  
Shu-Cherng Fang
Keyword(s):  
Linear Inequality ◽  
Optimal Solution ◽  
Path Following ◽  
Inequality Constraints ◽  
Computational Effort ◽  
Optimization Techniques ◽  
Linear Program ◽  
Programming Approach ◽  
Linear Programs ◽  
Dual Program

A dual convex programming approach to solving linear programs with inequality constraints through entropic perturbation is derived. The amount of perturbation required depends on the desired accuracy of the optimum. The dual program contains only non-positivity constraints. Anϵ-optimal solution to the linear program can be obtained effortlessly from the optimal solution of the dual program. Since cross-entropy minimization subject to linear inequality constraints is a special case of the perturbed linear program, the duality result becomes readily applicable. Many standard constrained optimization techniques can be specialized to solve the dual program. Such specializations, made possible by the simplicity of the constraints, significantly reduce the computational effort usually incurred by these methods. Immediate applications of the theory developed include an entropic path-following approach to solving linear semi-infinite programs with an infinite number of inequality constraints and the widely used entropy optimization models with linear inequality and/or equality constraints.

On Sufficient Global Optimality Conditions for Bivalent Quadratic Programs with Quadratic Constraints

2015 ◽  
Vol 32 (04) ◽  
pp. 1550025
Author(s):  
Yu-Jun Gong ◽  
Yong Xia
Keyword(s):  
Optimization Problem ◽  
Sufficient Conditions ◽  
Global Optimality ◽  
Dual Model ◽  
Zero Duality Gap ◽  
Quadratic Constraints ◽  
Quadratic Optimization Problem ◽  
Lagrangian Dual ◽  
Dual Bound ◽  
Global Optimality Condition

We show the recent sufficient global optimality condition for the quadratic constrained bivalent quadratic optimization problem is equivalent to verify the zero duality gap. Then, based on the optimal parametric Lagrangian dual model, we establish improved sufficient conditions by strengthening the dual bound.

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