Quadratically constrained quadratic program

In <a href="/facts/Mathematical_optimization/oRn8Iv5I">mathematical optimization</a>, a quadratically constrained quadratic program (QCQP) is an <a href="/facts/Optimization_problem/QzwJfFEE">optimization problem</a> in which both the <a href="/facts/Objective_function/xv5ozuhl">objective function</a> and the <a href="/facts/Constrained_optimization/tJuCmQ4b">constraints</a> are <a href="/facts/Quadratic_function/mKxHIC3U">quadratic functions</a>. It has the form

minimize
                
              
              
              
                
                  
                    
                      1
                      2
                    
                  
                
                
                  x
                  
                    
                      T
                    
                  
                
                
                  P
                  
                    0
                  
                
                x
                +
                
                  q
                  
                    0
                  
                  
                    
                      T
                    
                  
                
                x
              
            
            
              
              
                
                  subject to
                
              
              
              
                
                  
                    
                      1
                      2
                    
                  
                
                
                  x
                  
                    
                      T
                    
                  
                
                
                  P
                  
                    i
                  
                
                x
                +
                
                  q
                  
                    i
                  
                  
                    
                      T
                    
                  
                
                x
                +
                
                  r
                  
                    i
                  
                
                ≤
                0
                
                
                  for 
                
                i
                =
                1
                ,
                …
                ,
                m
                ,
              
            
            
              
              
              
              
                A
                x
                =
                b
                ,
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&{\text{minimize}}&&{\tfrac {1}{2}}x^{\mathrm {T} }P_{0}x+q_{0}^{\mathrm {T} }x\\&{\text{subject to}}&&{\tfrac {1}{2}}x^{\mathrm {T} }P_{i}x+q_{i}^{\mathrm {T} }x+r_{i}\leq 0\quad {\text{for }}i=1,\dots ,m,\\&&&Ax=b,\end{aligned}}}

where P0, ..., Pm are n-by-n matrices and x ∈ Rn is the optimization variable.
If P0, ..., Pm are all <a href="/facts/Positive-definite_matrix/Hbr6AuPS">positive semidefinite</a>, then the problem is <a href="/facts/Convex_set/vdAuJRJl">convex</a>. If these matrices are neither positive nor negative semidefinite, the problem is non-convex. If P1, ... ,Pm are all zero, then the constraints are in fact <a href="/facts/Linear_map/Q1PBKNVV">linear</a> and the problem is a <a href="/facts/Quadratic_program/ct4RIUgs">quadratic program</a>.

Quadratically constrained quadratic program open-in-new

Quadratically constrained quadratic program