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No, this is not true (unless P=NP). There are examples of convex optimization problems which are NP-hard. Several NP-hard combinatorial optimization problems can be encoded as convex optimization prob …

An obvious sufficient condition is that the SDP in question has a rank-$1$ optimal solution. Indeed, the SDP provides you with an upper bound on the MAXCUT value, and then you pay the price of $\alpha …

If $g_j$ is SOS, i.e. $g_j=\sum_k h_k^2$, then $g_j(x)\leq 0$ iff $g_j(x)=0$, i.e. $h_1(x)=h_2(x)=\dots =0$. So this is a general case, although with equality constraints only. A more interesting que …

it appears that one can eliminate one block of quantified variables in such a quadratic case faster, for fixed $r$, than in general, see e.g. https://arxiv.org/abs/0708.3522, but the result will have …

The best known general result is due to Eva Tardos' "A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs", published in 1986. Basically, it says that only bitsizes of coefficients …

While $f(x)=\sum_i \sum_j (x_i-x_j)^2$ is not quite what you might want, as it would favour vectors with small discrepancy in the entries, it's at least a very nice convex function to minimise, and co …

If you like, you might look at cones of sums of squares of polynomials (cones of PSD matrices are the same thing as cones of sums of squares of linear polynomials). This is the starting point of the m …

A usual trigonometric substitution $x_j:=\cos\phi_k +i\sin\phi_k$, $\phi:=(\phi_1,\dots,\phi_n)$ might tell something. After some straightforward manipulations your minimisation problem becomes $$\min …

It's a long comment, not an answer. Already the well-known SDR cone $\Sigma_{n,d}$ of sums of squares of homogeneous $n$-variate degreed $d$ polynomials has an interesting dual (also SDR), described …

Quadratic optimization subject to fixed number of quadratic constraints is "easy", even without any convexity assumptions. The algorithms are polynomial-time, but in practice quite hard to implement e …