How to express log(1/x + 1/y)?

Dear CVX forum,

I am working with a problem involving \log \sum_j \frac{a_j}{x_j}, where a_j's are non-negative constants and x_j \ge 0, \forall j are real-valued variables.

The expression is convex w.r.t. \{x_j\}_{j \in [J]}, but it violates DCP ruleset and I haven’t find a way using built-in functions to construct a valid expression. The tricks here for \log(1 + 1/x) does not work for multiple variables.

I was wondering if anyone could provide some suggestions on this problem, which may be helpful for optimization in index-modulated wireless systems.

Thank you in advance and wish you a lovely Easter.

Best regards,

Use log_sum_exp in the first approach in section 5.2.7 Log-sum-inv of .

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Thank you so much Mark! The MOSEK manual is very helpful.